ARITHMETIC OF L-FUNCTIONS

CHAO LI

NOTES TAKEN BY PAK-HIN LEE

Abstract. These are notes I took for Chao Li’s course on the arithmetic of L-functionsoffered at Columbia University in Fall 2018 (MATH GR8674: Topics in Number Theory).

WARNING: I am unable to commit to editing these notes outside of lecture time, so theyare likely riddled with mistakes and poorly formatted.

Contents

1. Lecture 1 (September 10, 2018) 41.1. Motivation: arithmetic of elliptic curves 41.2. Understanding ralg(E): BSD conjecture 41.3. Bridge: Selmer groups 61.4. Bloch–Kato conjecture for Rankin–Selberg motives 72. Lecture 2 (September 12, 2018) 72.1. L-functions of elliptic curves 72.2. L-functions of modular forms 92.3. Proofs of analytic continuation and functional equation 103. Lecture 3 (September 17, 2018) 113.1. Rank part of BSD 113.2. Computing the leading term L(r)(E, 1) 123.3. Rank zero: What is L(E, 1)? 144. Lecture 4 (September 19, 2018) 15

4.1. Rationality of L(E,1)Ω(E)

15

4.2. What is L(E,1)Ω(E)

? 17

4.3. Full BSD conjecture 185. Lecture 5 (September 24, 2018) 195.1. Full BSD conjecture 195.2. Tate–Shafarevich group 205.3. Tamagawa numbers 216. Lecture 6 (September 26, 2018) 226.1. Bloch’s reformulation of BSD formula 226.2. p-Selmer groups 237. Lecture 7 (October 8, 2018) 257.1. p∞-Selmer group 257.2. Tate’s local duality 26

Last updated: December 14, 2018. Please send corrections and comments to phlee@math.columbia.edu.1

7.3. Local conditions at good primes 277.4. Kolyvagin’s method 288. Lecture 8 (October 10, 2018) 288.1. Kolyvagin’s method 288.2. Heegner points on X0(N) 298.3. Heegner points on elliptic curves 309. Lecture 9 (October 15, 2018) 319.1. Gross–Zagier formula 319.2. Back to E/Q 329.3. Gross–Zagier formula for Shimura curves 3310. Lecture 10 (October 17, 2018) 3410.1. Gross–Zagier formula for Shimura curves 3410.2. Waldspurger’s formula 3511. Lecture 11 (October 22, 2018) 3711.1. Examples of Waldspurger’s formula 3711.2. Back to Kolyvagin 3812. Lecture 12 (October 24, 2018) 3912.1. Geometric interpretation of Ribet’s theorem: Ihara’s lemma 4012.2. Geometry of modular curves in characteristic p 4113. Lecture 13 (October 29, 2018) 4213.1. The supersingular locus 4213.2. Geometry of Shimura curves 4314. Lecture 14 (October 31, 2018) 4514.1. Inertia action on cohomology 4514.2. Nearby cycle sheaves and monodromy filtration 4614.3. Rapoport–Zink weight spectral sequence 4715. Lecture 15 (November 7, 2018) 4816. Lecture 16 (November 12, 2018) 4817. Lecture 17 (November 14, 2018) 4818. Lecture 18 (November 19, 2018) 4818.0. Rankin–Selberg L-functions for GLn ×GLn−1 4818.1. Global zeta integrals 4918.2. Relation between Z(f, ϕ, s) and L(s, π × π′) 5218.3. Global L-functions 5219. Lecture 19 (November 26, 2018) 5319.1. General Waldspurger formula 5319.2. Gan–Gross–Prasad conjectures 5520. Lecture 20 (November 28, 2018) 5520.1. Bloch–Kato Selmer groups 5520.2. Bloch–Kato conjecture 5820.3. Rankin–Selberg motives 5821. Lecture 21 (December 3, 2018) 5821.1. BBK for Rankin–Selberg motives 5821.2. Rank one case 6022. Lecture 22 (December 5, 2018) 62

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22.1. Proof strategies 6222.2. Geometry of unitary Shimura varieties at good reduction 6322.3. Tate’s conjecture 6422.4. Second reciprocity 6423. Lecture 23 (December 10, 2018) 6523.1. Tate conjecture via geometric Satake 6523.2. Geometric Satake 6623.3. Spectral operators 6623.4. Computing intersection numbers 68

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1. Lecture 1 (September 10, 2018)

1.1. Motivation: arithmetic of elliptic curves. Let us begin with some motivation bystudying the arithmetic of elliptic curves. Consider an elliptic curve E : y2 = x3 + Ax + B,where A,B ∈ Q.

Example 1.1. The elliptic curve y2 = x3−x intersects with the x-axis at (−1, 0), (0, 0) and(1, 0).

Given two points P , Q on E, the straight line through P and Q intersects E at a thirdpoint R. Declaring that P + Q + R = 0 defines an abelian group law on E and on the setof rational points E(Q). There are some degenerate cases; for example, the tangent linethrough P ′ ∈ E defines the relation 2P ′ +R′ = 0.

A fundamental result in the arithmetic of elliptic curves is that the group of rational pointscannot be too huge.

Theorem 1.2 (Mordell). E(Q) is finitely generated.

Definition 1.3. The algebraic rank of E is defined as ralg(E) := rankE(Q) ∈ Z≥0.

In particular, ralg(E) = 0 if and only if E(Q) is finite.

Example 1.4. For E : y2 = x3 − x, E(Q) = Z/2 × Z/2 is generated by the two 2-torsionpoints (0, 0) and (1, 0). Thus ralg(E) = 0.

Example 1.5. For E : y2 = x3 − 25x, one can check that P = (−4, 6) ∈ E(Q), and2P = (1681

144,−62279

1728) ∈ E(Q). In fact, E(Q) ' Z×Z/2×Z/2, with generators (−4, 6), (0, 0),

(5, 0) respectively, and thus ralg(E) = 1.

The greatest mystery in the study of elliptic curves is the following question. Given E/Q,how do we determine ralg(E)? No algorithm exists yet! In fact, we have very little knowledge;we don’t even know which integers can show up as ralg(E).

To show how limited our knowledge is:

Example 1.6. Consider E(n) : y2 = x3 − n2x. The smallest n for each value of ralg(E(n)) is

ralg(E(n)) 0 1 2 3 4 5 6n 1 5 34 1254 29274 48272239 6611714866

but we don’t know of any n which gives ralg(E(n)) = 7.

1.2. Understanding ralg(E): BSD conjecture. In order to attack the problem of under-standing ralg(E), the deep connections with the arithmetic L-function come into play.

Analogous to the Riemann zeta function

ζ(s) =∑n≥1

1

ns=∏p

(1− p−s)−1,

we can define the L-function associated to E/Q:

L(E, s) =∑n≥1

anns

=∏p

Lp(E, s),

4

where the Euler factor is defined by

Lp(E, s) =

(1− app−s + p1−2s)−1 if p is good,

(1− app−s)−1 if p is bad,

and

ap =

p+ 1− |E(Fp)| if p is good,

±1 if p is multiplicative,

0 if p is additive.

Later we will give some motivation for defining the L-function this way. The whole point isthat these L-functions are much easier to study; for example, ap is a more tractable number,depending on the number of points of E mod p.L(E, s) converges when Re s > 3

2. Analogous to the Riemann zeta function, we expect

L(E, s) to admit analytic continuation and satisfy a functional equation. In fact, there isthe miraculous theorem:

Theorem 1.7 (Modularity theorem: Wiles, Taylor, BCDT). There exists a cuspidal eigen-form fE ∈ S2(NE) such that

fE(s) =∑n≥1

anqn

where the an’s are as before.

Example 1.8. Consider E : y2 = x3 − x, which has conductor NE = 32. Solving thisequation mod p, we see that

p 2 3 5 7 11 13 17 19 23 29 31ap 0 0 -2 0 0 6 2 0 0 -10 0

Notice that ap 6= 0 if and only if p ≡ 1 (mod 4). Under the modularity theorem, thecorresponding modular form is

fE = q∏n≥1

(1− q4n)2(1− q8n)2

= q − 2q5 − 3q9 + 6q13 + 2q17 − q25 − 10q29 + · · · .

The modularity theorem is truly miraculous; why does the number of solutions mod phave anything to do with modular forms? For us, we have the

Corollary 1.9. L(E, s) = L(fE, s). In particular, L(E, s) has analytic continuation to Cand satisfies a functional equation under s↔ 2− s.

Definition 1.10. The analytic rank of E is defined as ran(E) := ords=1 L(E, s) ∈ Z≥0.

Conjecture 1.11 (BSD). ralg(E) = ran(E).

Remark. BSD leads to an algorithm for computing ralg and ran.

Theorem 1.12 (Gross–Zagier, Kolyvagin, 1980’s).

(1) If ran(E) = 0, then ralg(E) = 0.(2) If ran(E) = 1, then ralg(E) = 1.

But little is known in the higher rank case!5

1.3. Bridge: Selmer groups. Let us outline how this theorem is proved.

E(Q) oovery far

//

(∗)

L(E, s)

(∗∗)

Selp∞(E)

where the connection (∗) is made by p∞-descent, and (∗∗) is by explicit reciprocity laws.Recall the pn-descent sequence

0→ E(Q)/pnE(Q)→ Selpn(E)→X(E)[pn]→ 0

where Selpn(E) is a finite group and X(E) is the Tate–Shafarevich group.

Conjecture 1.13. X(E) is finite.

Taking inverse limit over n and tensoring with Qp, we get an injection

E(Q)⊗Qp → Selp∞(E)

which is an isomorphism if X(E) is finite. So

finiteness of X(E) =⇒ ralg(E) = dimQp Selp∞(E).

dimQp Selp∞(E) is called the Selmer rank of E.In summary, the algebraic object Selp∞(E) gives an upper bound for ralg(E), which is

sharp if X(E) is finite. Moreover, Selp∞(E) is easier to understand due to its local nature.Strategy for proving Theorem 1.12:

(1) ran = 0 =⇒ Selp∞(E) = 0.(2) ran = 1 =⇒ dim Selp∞(E) ≤ 1. Then show ralg(E) ≥ 1 by finding a point of infinite

order.

The connection between ran and Selmer groups is obtained by bounding Selmer groups fromabove using Euler systems (due to Kolyvagin), and the explicit point construction is byconsidering Heegner points (due to Gross–Zagier).

The key in bounding Selmer groups is to construct Galois cohomology classes with con-trolled local ramification. There are currently three ways to do this:

• Kolyvagin: Heegner points in ring class fields• Bertolini–Darmon: Heegner points on different Shimura curves• Kato: Kato classes in cyclotomic fields (in case (1) only)

We will focus on the second approach by Bertolini–Darmon, and generalize to more generalmotives.

Let me briefly indicate where the ramification comes from in each case:

• Kolyvagin: tamely ramified extensions• Bertolini–Darmon: ramified groups• Kato: wildly ramified extensions

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1.4. Bloch–Kato conjecture for Rankin–Selberg motives. The second part of thecourse will be on some recent results on Bloch–Kato conjecture for certain Rankin–Selbergmotives.

For elliptic curves, the Bloch–Kato conjecture takes the following form:

Conjecture 1.14 (Bloch–Kato). dim Selp∞(E) = ran(E).

This is exactly as predicted by BSD and the finiteness of X.Bloch–Kato gives an alternative definition of Selmer groups H1

f (Q, ρE) where ρE : GQ →GL2(Qp) is the Galois representation on the rational Tate module Tp(E)⊗Qp. The Bloch–Kato Selmer groups H1

f are Qp-vector spaces associated to arbitrary p-adic Galois represen-tations coming from geometry, which coincide with the usual Selmer groups in the case ofelliptic curves. Now we can generalize the earlier picture

fE ∈ S2(N)

uu &&

H1f (Q, ρE) ∼= Selp∞(E) L(fE, s)

Let Π be a conjugate self-dual cohomological cuspidal automorphic representation onGLn(AK) where K is a CM quadratic extension of a totally real field F . By the globalLanglands correspondence (due to many people, e.g. Chenevier–Harris), we can associate toΠ a Galois representation ρΠ : GK → GLn(Qp).

Now consider Π on GLn(AK) and Π′ on GLn+1(AK), which give the Rankin–Selbergmotive H1

f (K, ρ∨Π ⊗ ρ∨Π′) and the Rankin–Selberg L-function L(Π × Π′, s). The Bloch-Katoconjecture in this case asserts that

Conjecture 1.15 (Bloch–Kato). dimH1f (K, ρ∨Π ⊗ ρ∨Π′) = ords= 1

2L(Π× Π′, s).

Theorem 1.16 (Liu–Tian–Xiao–Zhang–Zhu). Assume Π and Π′ have trivial infinitesimalcharacter and p satisfies certain assumptions.

(1) If ords= 12L(Π× Π′, s) = 0, then H1

f (K, ρ∨Π ⊗ ρ∨Π′) = 0.

(2) Assume clGGP(Π×Π′) ∈ H1f (K, ρ∨Π⊗ρ∨Π′) is nonzero. Then dimH1

f (K, ρ∨Π⊗ρ∨Π′) = 1.

Remark. By the Beilinson conjecture, clGGP(Π× Π′) 6= 0 ⇐⇒ ords= 12L(Π× Π′, s) = 1.

2. Lecture 2 (September 12, 2018)

2.1. L-functions of elliptic curves. Recall: for an elliptic curve E/Q, we define

L(E, s) =∑n≥1

anns

=∏p

Lp(E, s)

where the Euler factor at p is

Lp(E, s) =

(1− app−s + p · p−2s)−1 if p - N = NE,

(1± p−s)−1 if p ‖ N,1 if p2 | N.

7

A uniform way to write this is as follows: define χ(p) =

1 if p - N,0 if p | N,

, so then

Lp(E, s) = (1− app−s + χ(p)p · p−2s)−1.

Some easy consequences are:

(1) a1 = 1.(2) amn = aman if (m,n) = 1.(3) apr = apapr−1 − χ(p)papr−2 if r ≥ 2.

This exactly matches the relations for Hecke operators Tn.

Remark. L(E, s) is an example of a “motivic L-function”, i.e., one associated to Galoisrepresentations coming from geometry. Recall the `-adic Tate module T`E = lim←−nE[`n],which is a Z`-module of rank 2. The action of GQ on V`E = T`E ⊗ Q` gives rise to a2-dimensional Galois representation

ρE : GQ → Aut(V`E) ' GL2(Q`).

Then for p 6= `,

Lp(E, s) = det(1− p−s Frp | (V`E)Ip

)−1

where Ip ⊂ GQp is the inertia subgroup. Notice that

dim(V`E)Ip =

2 if p - N,1 if p ‖ N,0 if p2 | N.

In particular, ap = tr(Frp | (V`E)Ip).

Proposition 2.1. L(E, s) converges when Re s > 32.

Proof. By counting the number of solutions to y2 = x3 + Ax + B, we have the trivialbound |E(Fp)| ≤ 2p + 1. However, 50% chance of being a square in Fp gives the heuristic

|E(Fp)|?∼ p+ 1. This is made precise by

Theorem 2.2 (Hasse). |p+ 1− |E(Fp)|| ≤ 2√p.

This implies ∣∣∣anns

∣∣∣ = O

(1

nRe s− 12

)and so

∑anns

converges if Re s− 12> 1.

Remark. Hasse’s theorem can be seen in a high-brow manner using the Weil conjectures,since ap = α + β with |α| = |β| = p1/2.

Expected properties of L(E, s) (more generally any motivic L-function):

(1) Euler product.(2) L(E, s) has analytic continuation to all of C.(3) L(E, s) has a functional equation.

8

(2) and (3) are more difficult and is one of the motivations of the Langlands program.

L(E, s)motivic L-function

oo //L(fE, s)

automorphic L-function

Automorphic L-functions are associated to modular forms or automorphic forms, for which(2) and (3) are easier.

2.2. L-functions of modular forms. Let Γ0(N) =

(a bc d

): c ≡ 0 (mod N)

⊆ SL2(Z).

Let Sk(N) be the space of cusp forms of level Γ0(N) and weight k. Recall f ∈ Sk(N) if:

(1) f : H → C is holomorphic.

(2) f(γτ) = (cτ + d)kf(τ) if γ =

(a bc d

)∈ Γ = Γ0(N).

(3) f is holomorphic and vanishes at all cusps.

Every f ∈ Sk(N) has a Fourier expansion f(τ) =∑

n≥1 an(f)qn.

Definition 2.3. The L-function of a cusp form f ∈ Sk(N) is defined as

L(f, s) :=∑n≥1

anns,

where an = an(f).

Remark. In this generality, L(f, s) may not have an Euler product. It has an Euler productif and only if f is eigenform for all Tn’s.

Proposition 2.4. L(f, s) converges when Re s > k2

+ 1. In fact, |an| = O(nk/2).

Remark.

• This is the trivial bound for E/Q (for weight k = 2).

• If f is an eigenform, there is a sharper bound |ap| = O(pk−12 ) (due to Deligne).

Proof. Notice

an =

∫ 1

0

e−2πin(x+iy)f(x+ iy) dx

for any y. Taking y = 1n, we have

|an| = e2π

∣∣∣∣∫ 1

0

f

(x+ i · 1

n

)dx

∣∣∣∣ .On the other hand, |f(τ)|(im τ)

k2 is invariant under Γ. Since f is a cusp form, the function

|f(τ)|(im τ)k2 is extended to Γ\H∗, so it is bounded on H. This implies |f(x+iy)| = O(y−

k2 ).

For y = 1n, we have |f(x+ i · 1

n)| = O(n

k2 ) and so an = O(n

k2 ).

The next goal is to prove (2) and (3) for L(f, s).9

2.3. Proofs of analytic continuation and functional equation. Starting point: L(f, s)has an explicit integral representation (Mellin transform of f).

Definition 2.5. The Mellin transform of f is

g(s) =

∫ ∞0

f(it)tsdt

t.

Proposition 2.6. g(s) = (2π)−sΓ(s)L(f, s) when Re s > k2

+ 1.

Proof. Recall Γ(s) =∫∞

0e−tts dt

t. Changing t by 2πnt,

Γ(s) = (2πn)s∫ ∞

0

e−2πnttsdt

t,

so

n−s = (2π)sΓ(s)−1

∫ ∞0

e−2πnttsdt

t.

Summing over n gives

L(f, s) =∑n≥1

anns

= (2π)sΓ(s)−1∑n≥1

an

∫ ∞0

e−2πnttsdt

t

= (2π)sΓ(s)−1

∫ ∞0

∑n≥1

ane−2πntts

dt

t

= (2π)sΓ(s)−1

∫ ∞0

f(it)ts dt

= (2π)sΓ(s)−1g(s).

Therefore g(s) = (2π)−sΓ(s)L(f, s).

Definition 2.7. The complete L-function of f is

Λ(f, s) := Ns2 (2π)−sΓ(s)L(f, s) = N

s2 g(s).

To get a functional equation, we need some extra symmetry on f .

Definition 2.8 (Atkin–Lehner involution). Define an operator WN : Sk(N)→ Sk(N) by

(WNf)(τ) =

(i√Nτ

)kf

(− 1

Nτ

).

One can check that W 2N = 1.

Theorem 2.9 (Hecke). Let f ∈ Sk(N). Assume WNf = εf , where ε ∈ ±1. Then

(1) Λ(f, s) has analytic continuation to all of C.(2) There is a function equation

Λ(f, s) = εΛ(f, k − s).

ε is called the sign of the functional equation, and plays an important role in the BSDconjecture. Since N

s2 (2π)−sΓ(s) doesn’t have any zeros, we have the

Corollary 2.10. L(f, s) has analytic continuation to all of C.

Remark. This theorem applies to any newform f ∈ Sk(N), so Λ(f, s) = ±Λ(f, k − s).10

Proof of Theorem 2.9.

Λ(f, s) = Ns2

∫ ∞0

f(it)tsdt

t

=

∫ ∞0

f

(it√N

)tsdt

t

[by t 7→ t√

N

]=

∫ 1

0

f

(it√N

)tsdt

t︸ ︷︷ ︸(∗)

+

∫ ∞1

f

(it√N

)tsdt

t︸ ︷︷ ︸(∗∗)

.

The term (∗∗) clearly converges, whereas

(∗) =

∫ 1

0

(WNf)

(i√Nt

)ts−k

dt

t[by definition of WN ]

=

∫ ∞1

(WNf)

(it√N

)tk−s

dt

t[by t 7→ t−1].

Therefore,

Λ(f, s) =

∫ ∞1

f

(it√N

)tsdt

t+

∫ ∞1

(WNf)

(it√N

)tk−2 dt

t.

The theorem follows since WNf = εf .

3. Lecture 3 (September 17, 2018)

3.1. Rank part of BSD. Recall: For an elliptic curve E/Q,

L(E, s) =∏p

Lp(E, s).

By the modularity theorem, we have shown:

(1) L(E, s) has analytic continuation to C.(2) L(E, s) satisfies a functional equation under s↔ 2− s.

The complete L-function satisfies Λ(E, s) = εΛ(E, 2− s) where ε ∈ ±1.At the center of the functional equation s = 1, we have

ords=1 Λ(E, s) =

even if ε = +1,

odd if ε = −1.

Since ords=1 Λ(E, s) = ords=1 L(E, s),

ran(E) =

even if ε = +1,

odd if ε = −1.

Question. What is the relation between ran(E) and ralg(E)?

Heuristically, we can try plugging s = 1 into∏

p Lp(E, s), although this doesn’t converge!Note that

Lp(E, 1) = (1− ap · p−1 + χ(p)p · p−2)−1

11

=

(p− ap + χ(p)

p

)−1

=p

p− ap + χ(p)

=

p

p+1−ap if p is good,pp±1

if p is multiplicative,pp

if p is additive,

=p

|E(Fp)|

where E is the smooth part of the reduction of E mod p, so

L(E, 1)“=”∏p

p

|E(Fp)|.

If E(Q) has “more” points (i.e., ralg(E) is “large”), then |E(Fp)| is “large” for each p, andhence L(E, 1) becomes “more” zero (i.e., ran(E) is “large”).

In fact, Birch and Swinnerton-Dyer did numerical computation for y2 = x3 − n2x in1958. They computed ∏

p<X

|E(Fp)|p

“looks like”∼ c(logX)r

where r = ralg(E). This leads to:

Conjecture 3.1 (BSD: rank part). ran(E) = ralg(E).

However, all these are heuristics.

Question. How to compute ran(E)? L(r)(E, 1)?

3.2. Computing the leading term L(r)(E, 1). Recall:

Λ(f, s) =

∫ ∞1

f

(it√N

)(ts + εtk−s)

dt

t

where WNf = εf for the Atkin–Lehner involution. Let us apply this to f = fE ∈ S2(N).

3.2.1. Rank zero. For r = 0 (hence ε = +1),

Λ(E, 1) = 2

∫ ∞1

f

(it√N

)t · dt

t

= 2

∫ ∞1

∑n≥1

ane− 2πnt√

N dt

= 2∑n≥1

an

∫ ∞1

e− 2πnt√

N dt

= 2∑n≥1

√N

2π· ann· e−

2πn√N .

12

Then the incomplete L-function is

L(f, 1) =2π√N

Λ(f, 1)

= 2∑n≥1

anne− 2πn√

N .

This is a formula we can use to compute whether an elliptic curve has analytic rank 0.

Remark. L(f, 1) can be viewed as a “weighted sum” of the divergent series

L(f, 1)“=”∑ an

n.

3.2.2. Rank one. For r = 1 (hence ε = −1),

Λ′(f, 1) = 2

∫ ∞1

f

(it√N

)log t dt

= 2

∫ ∞1

∑n≥1

ane− 2πnt√

N log t dt

= 2∑n≥1

an

∫ ∞1

e− 2πnt√

N log t dt

= 2∑n≥1

√N

2π· ann

∫ ∞1

e− 2πnt√

Ndt

t,

where we have used integration by parts to obtain the exponential integral∫ ∞1

e−ct log t dt =1

c

∫ ∞1

e−ctdt

t.

Then the incomplete L-function has derivative

L′(f, 1) =2π√N

Λ′(f, 1)

= 2∑n≥1

ann

∫ ∞1

e− 2πnt√

Ndt

t.

3.2.3. General rank. More generally, for any r ≥ 1, we define

Er(x) :=1

(r − 1)!

∫ ∞1

e−xt(log r)r−1 dt

t.

Then the leading term is

L(r)(f, 1) = 2r!∑n≥1

annEr

(2πn√N

).

This shows that the analytic rank is easy to compute in practice.

Remark. ε is easy to compute. Breaking the integral into∫∞A

+∫∞A−1 gives

L(f, 1) =∑n≥1

ann

(e− 2πn

A√N + εe

− 2πnA√N

).

13

Now plugging in different values of A (e.g. A = 1 and A = 1.01) gives the only choice for ε.

3.3. Rank zero: What is L(E, 1)?

Example 3.2. Consider E : y2 = x3 − x, which has conductor N = 32 (and complexmultiplication by Q(i)). The corresponding modular form is

f = q∏n≥1

(1− q4n)2(1− q8n)2

= q − 2q5 − 3q9 + 6q13 + 2q17 + · · · .

In fact, ap 6= 0 ⇐⇒ p ≡ 1 (mod 4). Then

L(E, 1) = 2∑ an

ne− 2πn√

32 = 0.65551438837 · · · .

Example 3.3. Consider E : y2 = x3 + 1, which has conductor N = 36 (and complexmultiplication by Q(

√−3)). The corresponding modular form is

f = q∏n≥1

(1− q6n)4

= q − 4q7 + 2q13 + 8q19 − 5q25 + · · · .

In fact, ap 6= 0 ⇐⇒ p ≡ 1 (mod 3). Then

L(E, 1) = 2∑ an

ne−

πn3 = 0.701091052663 · · · .

Question. What are these transcendental numbers?

Answer. There are natural transcendental numbers associated to elliptic curves: ellipticintegrals, or periods of elliptic curves in modern language.

Definition 3.4. Suppose E has minimal Weierstrass equation

y2 + a1xy + a3y = x3 + a2x2 + a4x+ a6.

Define the Neron differential

ωE :=dx

2y + a1x+ a3

∈ H0(E,Ω1E)

and the Neron period is

Ω(E) :=

∫E(R)

ωE ∈ R.

Example 3.5. For E : y2 = x3 − x, this equation is already minimal and the Nerondifferential is

ωE =dx

2y=

dx

2√x3 − x

.

The real locus E(R) consists of two circles (including the point at infinity), which correspondto two homologous circles in the complex torus E(C). Then

Ω(E) =

∫E(R)

ωE = 2

∫circle

dx

2√x3 − x

= 4

∫ ∞1

dx

2√x3 − x

= 5.24411510858 · · · .

14

Observe the miraculous coincidence:L(E, 1)

Ω(E)≈ 1

8∈ Q!

Similarly,

Example 3.6. For E : y2 = x3 + 1, the Neron differential is ωE = dx2√x3+1

. The real locus

E(R) is only one circle, which corresponds to a circle in E(C). Then

Ω(E) =

∫ ∞−1

dx√x3 + 1

= 4.20654631 · · ·

andL(E, 1)

Ω(E)≈ 1

6∈ Q.

The upshot is the following

Conjecture 3.7. For every elliptic curve E/Q,

L(E, 1)

Ω(E)∈ Q.

We will prove this next time and explain what happens for L(r)(E, 1).

4. Lecture 4 (September 19, 2018)

4.1. Rationality of L(E,1)Ω(E)

. To prove L(E,1)Ω(E)

∈ Q, we need to relate L(E, 1) to a “period”.

In general, a period of an algebraic variety X/Q is a quantity of the form

period =

∫i-cycle

closed i-form,

where an i-cycle lives in Hi(X(C),Q) and a closed i-form lives in H idR(X/Q).

Example 4.1. For X = Gm and ω = dtt,∫S1

dt

t= 2πi.

Note that 2πi is the period of the exponential function (inverse of log).

Example 4.2. For an elliptic curve,∫E(R)

ωE = period of the Weierstrass ℘-function (inverse of elliptic integral).

Our goal is to show that L(E, 1) = L(fE, 1) is related to periods of the (compactified)modular curve X0(N) = Γ0(N)\H∗. Recall the Mellin transform interpretation of L(f, s):

L(f, s) =(2π)s

Γ(s)

∫ ∞0

f(it)tsdt

t.

Applying this to s = 1 (without worrying about convergence),

L(f, 1) = 2π

∫ ∞0

f(it) dtτ=it= −2πi

∫ i∞

0

f(τ) dτ.

15

Then ωf = 2πif(τ) dτ can be interpreted as a 1-form on X0(N), and

L(f, 1) = −∫ i∞

0

ωf

is an integral along the path 0→ i∞ on X0(N).Problem: 0→ i∞ is not a closed path. But it is not too far from being one, as shown in

the following

Theorem 4.3 (Manin–Drinfeld). Take any cusps α, β ∈ X0(N). Then α → β belongs toH1(X0(N)(C),Q), i.e., there exist closed loops γi ∈ H1(X0(N)(C),Z) and ci ∈ Q such that∫ β

α

ωf =∑i

ci

∫γi

ωf

for all f ∈ S2(N).

Remark. In other words, a rational multiple of L(f, 1) is indeed a period.

Remark. This theorem implies (in fact, is equivalent to saying) α − β gives a torsion pointon J0(N) = Jac(X0(N)).

Idea of proof. Use Hecke operators Tp for p - N :

Tp

∫ i∞

0

=

∫ i∞

0

+

p−1∑k=0

∫ i∞

kp

= (1 + p)

∫ i∞

0

−p−1∑k=0

∫ 0

kp

.

Note that for p - N , kp→ 0 is a closed loop on X0(N). Then

(1 + p− Tp)∫ i∞

0

=∑∫

γi

with 1 + p − Tp acting as an integer (the number of points mod p). It remains to choose psuch that this is nonzero, which is possible because a modular form with Tp acting by 1 + pfor all p must be an Eisenstein series.

Now we are ready to prove the

Theorem 4.4 (Birch).L(E, 1)

Ω(E)∈ Q.

Proof. The modularity theorem gives a nontrivial map ϕ : X0(N) → E (defined over Q)such that ϕ∗(ωE) = c · 2πif(τ) dτ for some c ∈ Q×. Then

L(f, 1) ∼Q×

period of 2πif(τ) dτ ,

Ω(E) =period of ωE.

Both are real periods, so we conclude L(f, 1) ∼Q×

Ω(E).

Conjecture 4.5. c = 1.16

4.2. What is L(E,1)Ω(E)

? More generally, what is L(r)(E,1)Ω(E)

for r = ran(E)?

Example 4.6. Consider E : y2 = x3 − 25x, with N = 800 = 32 · 25, r = 1 and E(Q) ∼=Z × Z/2 × Z/2 generated by (−4, 6), (0, 0), (5, 0) respectively. The corresponding modularform is

f = q − 3q9 − 6q13 − 2q17 + · · · .We can use the formula from last time to compute that

L′(f, 1) = 2∑ an

nE1

(2πn√800

)= 2.22737037954 · · · .

However,

Ω(E) = 2

∫ ∞5

dx√x3 − 25x

= 2.34523957 · · ·

andL′(f, 1)

Ω(E)= 0.949741 · · · /∈ Q!

Of course, Theorem 4.4 is trivially true, since L(f,1)Ω(E)

= 0.

Idea: Use arithmetic complexity of the rational points of infinite order.

Definition 4.7. For x = ab∈ Q× with (a, b) = 1, define its height by

h(x) := log max|a|, |b|.

Definition 4.8. For P ∈ E(Q), define its naive height by

h(P ) :=

h(x(P )) if x(P ) 6= 0,

−∞ if x(P ) = 0.

Problem: This depends on the Weierstrass equation of E.

Definition 4.9 (Tate). Define the canonical height (or Neron–Tate height) by

h(P ) := limn→∞

h([2n]P )

4n∈ R≥0.

Example 4.10. Let us return to the previous example E : y2 = x3 − 25x. Then

P = (−4, 6) h(P ) = 1.856786 · · ·2P = (1681

144, · · · ) h(2P )

4= 1.8778407 · · ·

4P = (11183412793921223416132416

, · · · ) h(4P )16

= 1.89946579 · · ·In fact, h(P ) = 1.89948217253 · · · . Observe that

L′(f, 1)

Ω(E)h(P )=

1

2∈ Q!

Remark. h(P ) does not depend on the Weierstrass equation and is uniquely characterizedby the following:

(1) h : E(Q)→ R≥0.

(2) h(2P ) = 4h(P ).

(3) h(P )− h(P ) is bounded for P ∈ E(Q).17

Definition 4.11. Let r ≥ 2. Define the Neron–Tate height pairing 〈−,−〉 : E(Q)×E(Q)→R≥0 by

〈P,Q〉 :=1

2

(h(P +Q)− h(P )− h(Q)

).

In particular, 〈P, P 〉 = h(P ).

Definition 4.12. The regulator of E is defined to be

R(E) := det (〈Pi, Pj〉r×r) ,where Pi1≤i≤r are generators of E(Q)/E(Q)tors.

Remark.

(1) h(P ) = 0 ⇐⇒ P ∈ E(Q)tors, so 〈−,−〉 is perfect on E(Q)/E(Q)tors and R(E) 6= 0.(2) When r = 0, R(E) = 1.

(3) When r = 1, R(E) = h(P ).

Conjecture 4.13.L(r)(E, 1)

Ω(E)R(E)∈ Q×.

This is the first step towards the full BSD conjecture

Remark.

• When r = 0, this is true (proved).• When r = 1, this is true by the r = 0 case and Gross–Zagier formula.• When r ≥ 2, no example is known! But there is enormous numerical evidence.

4.3. Full BSD conjecture. Our goal is to find a formula forL(r)(E, 1)

Ω(E)R(E). Recall the three

examples we did earlier:

E N r L(r)(E,1)Ω(E)R(E)

y2 = x3 − x 32 0 18

y2 = x3 + 1 36 0 16

y2 = x3 − 25x 800 1 12

Idea: Instead of looking at∫E(R)

ωE only, we should look at∫E(Qp)

|ωE|.

Fact. ∫E(Qp)

|ωE| = [E(Qp) : E0(Qp)] ·|E(Fp)|

p,

where E0(Qp) = P ∈ E(Qp) : P mod p ∈ E(Fp). (Recall that |E(Fp)|p

= Lp(E, 1)−1.)

Definition 4.14. The local Tamagawa number is defined to be

cp(E) := [E(Qp) : E0(Qp)].

Remark.

• cp(E) = Φ(Fp) where ΦFp is the component group scheme of the Neron model of Eat p.• cp = 1 if p - N .

18

This is easily computed using Tate’s algorithm.Now we can complete the table above:

E N r L(r)(E,1)Ω(E)R(E)

∏cp

L(r)(E,1)Ω(E)·

∏p cp·R(E)

E(Q)tors

y2 = x3 − x 32 0 18

c2 = 2 116

Z/2× Z/2y2 = x3 + 1 36 0 1

6c2 · c3 = 3 · 2 = 6 1

36Z/2× Z/3

y2 = x3 − 25x 800 1 12

c2 · c5 = 2 · 4 = 8 116

Z/2× Z/2

Next time, we will complete the full formula using the data we see.

5. Lecture 5 (September 24, 2018)

5.1. Full BSD conjecture. Last time we saw from a few examples with r = 0, 1 that

L(r)(E, 1)

Ω(E) ·∏

p cp(E) ·R(E)?=

1

|E(Q)tor|2.

Let’s rewrite this as

L(r)(E, 1)

r!?= Ω(E) ·

∏p

cp(E)︸ ︷︷ ︸period

· R(E)

|E(Q)tor|2︸ ︷︷ ︸regulator

.

Remark. R(E)|E(Q)tor|2 is more canonical: for any Λ ⊆ E(Q) of rank r, R(Λ)

[E(Q):Λ]2is independent of

the choice of Λ.

Notice the similarity with the class number formula: for any number field K/Q,

ress=1 ζK(s) =2r1(2π)r2

|dK |1/2︸ ︷︷ ︸period

· R(K)

|O×K,tor|︸ ︷︷ ︸regulator

· h(K)︸ ︷︷ ︸class number

.

Missing in the formula of BSD above is an analogue of h(K) = |Cl(K)|; this will be providedby the size of the Tate–Shafarevich group X(E).

Conjecture 5.1 (Full BSD conjecture). Let E/Q be an elliptic curve. Then

(1) (rank part) ralg(E) = ran(E).(2) (formula part)

L(r)(E, 1)

r!= Ω(E) ·

∏p

cp(E) · R(E)

|E(Q)tor|2· |X(E)|.

This can be seen as a vast generalization of the class formula formula.

Remark. (1) is known when ran(E) ≤ 1 (Gross–Zagier, Kolyvagin). (2) is known for many(but not all) E/Q with ran(E) ≤ 1; the main ingredient is Skinner–Urban’s work on theIwasawa main conjecture and Kato’s Euler systems.

19

5.2. Tate–Shafarevich group.

Definition 5.2. The Tate–Shafarevich group of E is defined to be

X(E) := ker

(H1(Q, E)→

∏v

H1(Qv, E)

).

In other words, an element of X(E) is represented by a genus 1 curve C/Q such thatJac(C) ' E and C(Qv) 6= ∅ for all places v of Q. X(E) measures the failure of local-globalprinciple for Q-points (for C).

Example 5.3 (Lind, 1940). The curve C : 2y2 = x4− 17 has points over all Qp and R, buthas no Q-points. This corresponds to the elliptic curve

E = Jac(C) : y2 = x3 + 17x

which has r = 0 and X(E) ' (Z/2)2.To check the BSD formula, we compute that

L(E, 1) = 3.6523 · · · ,Ω(E) = 1.82618 · · · ,∏

p cp(E) = c2 · c17 = 1 · 2 = 2,

E(Q)tor = Z/2Z,

and henceL(E, 1)

Ω(E)∏

p cp=

2

1 · 2= 1 and

|X(E)||E(Q)tor|2

=4

4= 1.

Example 5.4 (Selmer, 1951). The curve C : 3x3 + 4y3 + 5z3 = 0 has points over all Qp andR, but no Q-points. This corresponds to the elliptic curve

E = Jac(C) : x3 + y3 + 60z3 = 0,

(with Weierstrass equation y2 = x3 − 24300), which has X(E) = (Z/3)2.

Remark. In general, there is no algorithm to compute X(E). The full BSD formula helpsto compute |X(E)|.

Remark. To see why |X(E)| is an analogue of the class number, note that the class groupCl(K) can also be described in terms of Galois cohomology. Setting S = ResOK/Z Gm, wehave

Cl(K) ∼= ker

(H1(Q, S)→

∏v

H1(Qv, S)

)measuring the failure of any ideal that is locally principal but not globally principal. Weknow Cl(K) is finite (using geometry of numbers), but we don’t know the finiteness of X(E).

Conjecture 5.5. X(E) is finite.

Remark.

(1) This conjecture is true if ran(E) ≤ 1 (Gross–Zagier, Kolyvagin).(2) No example is known where ran(E) ≥ 2!

20

Remark (Cassels–Tate pairing). There exists an alternating pairing X(E)×X(E)→ Q/Zwhose kernal is the divisible part of X(E), so

X(E) is finite =⇒ X(E) ' G×G =⇒ |X(E)| is a square!

Remark. Delaunay (2001) developed Cohen–Lenstra heuristics for X(E) as “random” fi-nite abelian groups with nondegenerate alternating pairing (H, (−,−)) (with frequency

1Aut(H,(−,−))

). It predicts that the probability of p | |X(E)| is given by:

• r = 0:

f0(p) = 1−∏k≥1

(1− 1

p2k−1

).

• r = 1:

f1(p) = 1−∏k≥1

(1− 1

p2k

).

For example,

p f0(p) f1(p)2 0.58 0.313 0.36 0.125 0.21 0.04

None of this is support by actual data. However, we don’t even know if given p there existsE/Q such that p | |X(E)| (not to mention positive probability!).

Remark. It is known that for p = 2, 3, 5, 7, 13, X(E)[p∞] can be arbitrarily large. The proofuses the fact that X0(p) has genus 0, so p = 11 is not allowed here.

5.3. Tamagawa numbers. There is a reformulation of the BSD formula, due to Bloch, interms of Tamagawa numbers, which is similar to the case of linear algebraic groups.

Definition 5.6. Let G be a semisimple algebraic group over Q. Take a top differential ω(defined over Q) of G. Then |ω|v (volume form) gives a Haar measure µv on G(Qv). Thenthe Tamagawa measure is µ =

∏v µv on G(A), and the Tamagawa number of G is

τ(G) := µ(G(Q)\G(A)).

Theorem 5.7 (Weil’s conjecture). If G is simply-connected, then τ(G) = 1.

This is proved by Langlands, Lai, Kottwitz, etc.

Example 5.8. Let G = SL2/Q. Then

τ(G) = µ∞(SL2(Z)\SL2(R)) ·∏p

µp(SL2(Zp)).

But

µp(SL2(Zp)) =|SL2(Fp)|

p3= 1− p−2 = ζp(2)−1,

so

τ(G) = 1 =⇒ µ∞(SL2(Z)\SL2(R)) = ζ(2) =π2

6

=⇒ Vol(SL2(Z)\H) =π

3.

21

Example 5.9. Let G = SLn/Q. Then

τ(G) = 1 ⇐⇒ µ∞(SLn(Z)\SLn(R)) = ζ(2) · · · ζ(n)

⇐⇒ Vol(SLn(Z)\Hn) = n2n−1 · ζ(2) · · · ζ(n)

Vol(S1) · · ·Vol(Sn−1).

We can do this for E too (next time).

6. Lecture 6 (September 26, 2018)

6.1. Bloch’s reformulation of BSD formula. Last time we saw that for G = SLn, theTamagawa number is τ(G) = 1; this special case is due to Minkowski and Siegel usinggeometry of numbers, before the formulation of Weil’s conjecture for semisimple algebraicgroups. But for more general algebraic groups, τ(G) may not be well-defined.

Example 6.1. Let G = Gm. Then

µp(G(Zp)) =|Gm(Fp)|

p=p− 1

p= 1− 1

p.

Notice∏

p(1 −1p) doesn’t converge! To fix this, the idea is to regularize this product by

changing the local measure to be(1− 1

p

)−1

µp = ζp(1)µp,

and take

τ(G) =µ(G(Q)\G(A))

ζ∗(1),

where ζ∗(1) is the leading coefficient of ζ(s) at s = 1. Then τ(Gm) = 1.

Fro a general linear algebraic group, there exists an L-function L(G, s) such that for allp /∈ S (a finite set of bad primes),

|G(Fp)|pdimG

= Lp(G, 1)−1.

Example 6.2. Let G = Gm. Then L(G, s) = ζ(s).

Example 6.3. Let G = GLn. Then L(G, s) = ζ(s)ζ(s+ 1) · · · ζ(s+ n− 1).

Definition 6.4. For any algebraic group G, define the modified Tamagawa measure

µ∗ :=∏v/∈S

Lv(G, 1)µv ·∏v∈S

µv

and the Tamagawa number

τ(G) :=µ∗(G(Q)\G(A))

L∗S(G, 1).

Then we have the following generalization of Weil’s conjecture.

Theorem 6.5 (T. Ono, Oesterle, Kottwitz). For any connected linear algebraic group G/Q,

τ(G) =|Pic(G)tor||X(G)|

.

22

Here X(G) = ker (H1(Q, G)→∏

vH1(Qv, G)) is the Tate–Shafarevich set.

Remark. If G is linear, X(G) is known to be finite (Borel–Serre, 1966).

Remark. Suppose G is simply connected and semisimple. Then indeed |Pic(G)tor| = 1.The verification that |X(G)| = 1 is a hard theorem due to Harder (1965), Kneser (1966),and Chernousov (1989) for E8, which completed Kottwitz’s proof of the Tamagawa numberconjecture.

Bloch’s idea is to look at elliptic curves. Suppose ralg(E) = r. Take a generator Pi ∈ E(Q).

Using the fact that E ' E = Ext1(E,Gm), Pi gives rise to an extension of groups

0→ Gm → XPi → E → 0.

If Pi is a basis of E(Q), then

0→ Grm → X → E → 0.

Theorem 6.6 (Bloch). The Tamagawa number conjecture for X is equivalent to the BSDformula for E.

In fact, X(X) = X(E), and Bloch’s theorem gives an interpretation of the regulatorR(E) also as a volume.

6.2. p-Selmer groups. Selmer groups are more accessible than the Mordell–Weil groupE(Q) and the Tate–Shafarevich group X(E), and will be used to prove the BSD conjecture.

Start with

0→ E[p]→ Ep→ E → 0.

Taking H∗(Q,−), we get a long exact sequence

0→ E(Q)[p]→ E(Q)p→ E(Q)→ H1(Q, E[p])→ H1(Q, E)

p→ H1(Q, E)→ · · ·

and hence

0→ E(Q)

pE(Q)→ H1(Q, E[p])→ H1(Q, E)[p]→ 0.

We want to understand E(Q), but H1(Q, E[p]) is an infinite-dimensional Fp-space. Insteadwe look at the local picture:

0 //E(Q)

pE(Q)δ

//

H1(Q, E[p]) //

locv

H1(Q, E)[p] //

0

0 //∏v

E(Qv)

pE(Qv)

δv//∏v

H1(Qv, E[p]) //∏v

H1(Qv, E)[p] // 0

Here H1(Qv, E[p]) is finite for each v!

Definition 6.7. The p-Selmer group is

Selp(E) := x ∈ H1(Q, E[p]) : locv(x) ∈ im(δv) for all v.23

In other words, we have a pullback diagram

Selp(E) //

H1(Q, E[p])

∏v

E(Qv)

pE(Qv)//∏v

H1(Qv, E[p])

which induces a natural mapE(Q)

pE(Q)→ Selp(E).

By construction, we have a “p-descent exact sequence”

0→ E(Q)

pE(Q)→ Selp(E)→X(E)[p]→ 0.

Selp(E) is described locally, and hence more accessible than E(Q) and X(E) which containglobal information.

Theorem 6.8. Selp(E) is finite.

Idea of proof. Selp(E) ⊆ H1(Q, E[p]) can be thought of as a subspace “cut out” by localconditions im(δv) ⊆ H1(Qv, E[p]) for all v. View a class x ∈ H1(Q, E[p]) as a number fieldL/Q. Then the local condition at v /∈ S (finite set of bad primes) imply that L is unramifiedat v (control on the ramification). The theorem follows because there are only finitely manynumber fields L/Q with exponent p and unramified outside S!

Since dimFp E(Q)/pE(Q) = ralg(E) + dimFp E(Q)[p], we obtain the

Corollary 6.9 (p-descent).

dimFp Selp(E) = ralg(E) + dimFp E(Q)[p] + dimFp X(E)[p].

In particular, dimFp Selp(E) ≥ ralg(E) + dimFp E(Q)[p].

Remark. Selp(E) is most computable when p = 2.

Example 6.10 (Heath-Brown, 1994). For the family y2 = x3−n2x, Heath-Brown provedthat the probability that dim Sel2(E) = d+ dimE(Q)[2] = d+ 2 is given by

P (d) :=∏k≥0

(1 +

1

2k

)−1 d∏k=1

2

2k − 1.

For example,

d P (d)0 0.20971 0.41942 0.27963 0.07994 0.0107

24

Remark. Poonen–Rains developed Cohen–Lenstra heuristics (for all E/Q):

Prob(dim Selp = d)?=∏k≥0

(1 +

1

pk

)−1 d∏k=1

p

pk − 1. (6.1)

This has the following consequences:

(1)Average(|Selp(E)|) = p+ 1.

This is proved by Bhargava–Shankar for p = 2, 3, 5.(2)

Prob(ralg(E) ≥ 2) ≤ 1

p+

1

p2=⇒ Prob(ralg(E) ≥ 2) = 0.

This is consistent with Goldfeld’s conjecture; in fact, (6.1) implies that 50% of Ehave ralg = 0 and 50% have ralg = 1.

Remark.

(1) Selp(E) = 0 =⇒ ralg(E) = 0.(2) Assume X(E)[p∞] <∞. Then dim Selp(E) = 1 =⇒ ralg(E) = 1.

The implication in (2) is still open in general, but Skinner–Zhang proved many cases, andcombining with Bhargava they get that at least 20% of E/Q have ralg = 1.

Next time we will state a reformulation of BSD in terms of Selmer groups.

7. Lecture 7 (October 8, 2018)

Last time we defined the p-Selmer group Selp(E) ⊆ H1(Q, E[p]), which is a finite-dimensionalFp-vector space inside an infinite-dimensional global H1. It fits into a p-descent sequence

0→ E(Q)

pE(Q)→ Selp(E)→X(E)[p]→ 0.

Thus bounds on Selp(E) lead to bounds on the algebraic rank of E. However, one disadvan-tage with this sequence is that only the p-torsion of X(E) is present. Taking limit over allp-powers will allow us to see the p-primary part of X(E).

7.1. p∞-Selmer group. Consider the pn-descent sequence

0→ E(Q)

pnE(Q)→ Selpn(E)→X(E)[pn]→ 0.

Taking lim−→nwith respect to the maps Z/pn → Z/pn+1 and E[pn] → E[pn+1], we obtain

0→ E(Q)⊗ Qp

Zp

→ Selp∞(E)→X(E)[p∞]→ 0.

Definition 7.1. The p∞-Selmer group is Selp∞(E) := lim−→nSelpn(E).

Remark. We have an exact sequence

0→ E(Q)[p∞]

pnE(Q)[p∞]→ Selpn(E)→ Selp∞(E)[pn]→ 0.

In particular, if E(Q)[p] = 0, then Selp∞(E) ' Selp∞(E)[p∞].25

Notice that Selp∞(E) ' (Qp/Zp)r × (finite group).

Definition 7.2.

rp(E) := corank(Selp∞(E))

= number of copies ofQp

Zp

.

Corollary 7.3.

(1) rp(E) ≥ ralg(E).(2) rp(E) = ralg(E) ⇐⇒ X(E)[p∞] is finite.

In particular, BSD and the finiteness of X imply

Conjecture 7.4 (Bloch–Kato). rp(E) = ran(E) for all p.

Remark.

(1) rp(E) is more accessible than ralg(E) due to its local nature.(2) By a theorem of Bloch–Kato, Selp∞(E) can be defined in terms of the p-divisible

group E[p∞]/Q or Vp(E)/Q. Warning: In general Selp(E) may not only dependE[p].

Bloch–Kato generalizes this conjecture to any p-adic Galois representation V coming fromgeometry.

Our target theorem is

Theorem 7.5 (Gross–Zagier, Kolyvagin).

(1) ran(E) = 0 =⇒ rp(E) = 0.(2) ran(E) = 1 =⇒ rp(E) ≤ 1 and ralg(E) ≥ 1.

Remark. (1) and (2) imply that BSD and Bloch–Kato are true when ran(E) ≤ 1, and thatX(E)[p∞] is finite for all p.

The goal today is to illustrate how to bound Selp∞(E) from above in the simplest case.

7.2. Tate’s local duality. In rough terms, local duality allows us to quantify different localconditions in H1(Qv, E[p∞]).

Definition 7.6. Let M be a finite GQ-module (e.g. M = E[pn]). Define the Cartier dual ofM

M∨(1) := Homab.gp(M,Gm).

The natural pairing M ×M∨(1)→ Gm, together with the cup product, gives a pairing

〈−,−〉v : H i(Qv,M)×H2−i(Qv,M∨(1))→ H2(Qv,Gm) ' Q/Z.

Theorem 7.7 (Tate).

(1) 〈−,−〉v is perfect.(2) (Euler characteristic)

|H1(Qv,M)||H0(Qv,M)| · |H2(Qv,M)|

= |M ⊗ Zv|.

Apply this to M = E[pn]. Then M∨(1) = E[pn] and 〈−,−〉v is precisely the Weil pairing.26

Corollary 7.8. We have an isomorphism

H0(Qv, E[pn])∼→ H2(Qv, E[pn])∨

and a perfect pairingH1(Qv, E[pn])×H1(Qv, E[pn])→ Q/Z.

Corollary 7.9. If v 6= p, then

|H1(Qv, E[pn])| = |H0(Qv, E[pn])|2 = |E(Qv)[pn]|2.

Example 7.10. Let n = 1. Then

dimFp H1(Qv, E[p]) = 2 dimFp E(Qv)[p].

Definition 7.11. Define Lv ⊆ H1(Qv, E[pn]) to be the image of

E(Qv)

pnE(Qv)

δv→ H1(Qv, E[pn]).

ThenSelpn(E) = x ∈ H1(Q, E[pn]) : locv(x) ∈ Lv for all v

Theorem 7.12 (Tate). Lv is its own annihilator under 〈−,−〉v.It would be helpful if we could characterize the local conditions Lv without reference to

the local points E(Qv). This is possible in some cases.

7.3. Local conditions at good primes. The goal is to pin down Lv in terms of only E[pn].

7.3.1. Case v 6= p.

Definition 7.13 (“Unramified” or “finite” condition). Assume |M | is invertible in Zv. Define

H1f (Qv,M)

(= H1

ur(Qv,M))

:= H1(Fv,MIv) ⊆ H1(Qv,M),

i.e., the extension classes which are unramified.

The inflation-restriction sequence gives

0→ H1(Fv,MIv)→ H1(Qv,M)→ H1(Iv,M)Frobv=1 → 0.

This leads to a different kind of condition.

Definition 7.14 (“Singular” condition).

H1sing(Qv,M) := H1(Iv,M)Frobv=1

(=H1(Qv,M)

H1f (Qv,M)

).

Proposition 7.15. H1f (Qv,M) and H1

f (Qv,M∨(1)) are annihilators of each other under

〈−,−〉v.Corollary 7.16. H1

f (Qv,M)×H1sing(Qv,M

∨(1))→ Q/Z is a perfect pairing.

Apply this to M = E[pn].

Proposition 7.17. If v 6= p and E has good reduction at v, then

Lv = H1f (Qv, E[pn])

(= H1

et(Zv, E [pn])).

where E is the Neron model of E, so that E(Zv) = E(Qv).27

7.3.2. Case v = p. The situation is more involved, but the last proposition suggests thefollowing

Definition 7.18 (Flat condition). Let M be a finite flat group scheme over Zp. Define

H1f (Qp,M) := H1

fppf(Zp,M) ⊆ H1(Qp,M).

Proposition 7.19. H1f (Qp,M) and H1

f (Qp,M∨(1)) are annihilators of each other under〈−,−〉p.

Analogously with the case v 6= p, we have the

Proposition 7.20. If v = p and E has good reduction at p, then

Lv = H1f (Qp, E [pn]).

Remark. If n = 1 and p > 2, Raynaud shows that E[p] has a unique extension to Zp, soH1f (Qp, E [p]) is determined by E[p].

7.4. Kolyvagin’s method. The starting point is a simple application of global class fieldtheory: for s1, s2 ∈ H1(Q, E[p]), we have

∑v〈s1, s2〉v = 0. This condition can be used to

bound Selmer ranks.

Theorem 7.21. Let S = bad primes for E ∪ p. Assume we can construct c(`) ∈H1(Q, E[p]) for all ` /∈ S satisfying:

(1) For v /∈ S ∪ `, locv(c(`)) ∈ H1f (Qv, E[p]).

(2) For v = p, locp(c(`)) ∈ H1f (Qp, E[p]).

(3) For v ∈ S − p, locv(c(`)) = 0 (e.g. when E(Qv)[p] = 0).(4) For v = `, H1

f (Q`, E[p]) = Fp and 0 6= loc`(c(`)) ∈ H1sing(Q`, E[p]).

Then Selp(E) = 0.

Next time we will explain this.

8. Lecture 8 (October 10, 2018)

8.1. Kolyvagin’s method. Last time we defined, for good primes v, the unramified condi-tions

H1f (Qv, E[p]) ⊆ H1(Qv, E[p])

and for v 6= p, the singular conditions H1sing(Qv, E[p]). We have 〈H1

f , H1f 〉v = 0, and that

〈−,−〉v is perfect on H1f ×H1

sing.

Theorem 8.1. Let S = bad primes ∪ p. Assume we can construct classes c(`) ∈H1(Q, E[p]) for all ` /∈ S such that:

(1) For v /∈ S ∪ `, locv(c(`)) ∈ H1f (Qv, E[p]).

(2) For v = p, locp(c(`)) ∈ H1f (Qp, E[p]).

(3) For v ∈ S − p, locv(c(`)) = 0.(4) For v = `, H1

f (Qv, E[p]) = Fp and loc`(c(`)) 6= 0 ∈ H1sing(Qv, E[p]).

Then Selp(E) = 0.

28

Proof. Assume Selp(E) 6= 0, and take 0 6= s ∈ Selp(E). By Chebotarev density, we can find` /∈ S such that 0 6= loc`(s) ∈ H1

f (Q`, E[p]). Then

〈s, c(`)〉 =∑〈s, c(`)〉v

by (1)-(3)= 〈s, c(`)〉`

by (4)

6= 0

since s ∈ H1f and c(`) ∈ H1

sing. This is a contradiction, since 〈s, c(`)〉 = 0 by global class fieldtheory. Thus Selp(E) = 0.

Remark. The proof shows that we only need c(`) for “many” (a positive proportion of) ` /∈ S.

By the theorem, the key to bound Selmer groups is the construction of the classes c(`)`/∈Swith controlled ramification. Next, we will carry out the construction via Heegner points.

Remark. The existence of c(`) should be related to ran(E) = 0.

Selp(E) ooBSD/BK

//

dd

Kolyvagin’smethod

$$

L(E, s)::

explicit reciprocity law(Bertolini–Darmon)

zz

c(`)

8.2. Heegner points on X0(N). Recall the modular curve Y0(N) := Γ0(N)\H, where

Γ0(N) =

(a bc d

): N | c

⊆ SL2(Z), and the compactified modular curve X0(N) :=

Γ0(N)\H∗. More precisely, what we are defining here are the complex points Y0(N)(C)and X0(N)(C).

In the simplest case, Y0(1) = SL2(Z)\H. For τ ∈ H, consider the elliptic curve Eτ :=C/Z + Zτ . It is easy to check that Eτ ' Eτ ′ if and only if τ and τ ′ are in the sameSL2(Z)-orbit, so

Y0(1)(C) = elliptic curves E/C.

To describe Y0(N), consider pairs(Eτ , Cτ =

1NZ+Zτ

Z+Zτ' Z/N

). Analogously as above,

(Eτ , Cτ ) ' (Eτ ′ , Cτ ′) if and only if τ and τ ′ are in the same Γ0(N)-orbit, and hence

Y0(N)(C) = (E,C) : E elliptic curve over C, C ⊆ E cyclic group of order N= E → E ′ = E/C: cyclic N -isogeny.

This moduli interpretation gives a model of X0(N) (and Y0(N)) over Q.We want to construct points on X0(N) and use the modular parametrization X0(N)/Q →

E/Q. However, X0(N)(Q) tends to be small (as a curve of genus at least 2 except for smallvalue of N). To overcome this problem, we instead construct algebraic points on X0(N)lying in small degree number fields.

As a naive try, we consider the uniformization H∗ → X0(N)(C). Although H∗ containsmany algebraic points, this map is highly transcendental and will not send algebraic pointsto algebraic points! The miracle, though, is that this may still work in special cases, namelyfor imaginary quadratic points. To see this, we use the moduli interpretation. Recall ellipticcurves E over C can be divided into two possibilities:

(1) End(E) = Z.29

(2) End(E) ∼= O ⊆ OK , where K = Q(√−|dK |) is an imaginary quadratic field and

O ⊆ OK is a finite index subring.

Definition 8.2. An elliptic curve E in case (2) is said to have complex multiplication (CM).

Example 8.3. The elliptic curve E : y2 = x3 + nx has CM by O = Z[i], where

[i](x, y) = (−x, iy).

Example 8.4. The elliptic curve E : y2 = x3 + n has CM by O = Z[ζ3], where

[ζ3](x, y) = (ζ3x, y).

The theory of complex multiplication tells us that CM elliptic curves are defined overnumber fields. Recall that for a number field K, the Hilbert class field HK is the maximalunramified abelian extension of K, with Gal(HK/K) ' Cl(K).

Theorem 8.5 (Main Theorem of CM). There is a bijection

elliptic curves with CM by OK∼→ C/a : a ∈ Cl(K).

Moreover, each such elliptic curve can be defined over the Hilbert class field HK.

Definition 8.6. A Heegner point is a point xK = (Eφ→ E ′) ∈ X0(N)(C), where φ is a

cyclic N -isogeny, such that End(E) = End(E ′) = OK .

We know that xK ∈ X0(N)(HK).

Remark. A Heegner point xK ∈ X0(N) exists ⇐⇒ there exists a, b ∈ Cl(K) such thatC/a→ C/b is a cyclic N -isogeny ⇐⇒ b/a ' OK/ab−1 is isomorphic to Z/N ⇐⇒ thereexists an ideal N ⊆ OK such that OK/N ' Z/N .

Definition 8.7. We say that K satisfies the Heegner hypothesis for X0(N) if every primedividing N splits in K.

If this is true, then taking a prime p above every p | N and N =∏

pordp(N) gives O/N =Z/N , and so xK exists. By Chebotarev density, there are infinitely many K satisfying theHeegner hypothesis for a given N .

8.3. Heegner points on elliptic curves.

Definition 8.8. Fix a modular parametrization ϕ : X0(N) → E sending ∞ ∈ X0(N) to0 ∈ E. Define a Heegner point on E to be

yK :=∑

σ∈Gal(HK/K)

σ(ϕ(xK)) ∈ E(K).

It is not necessary to further trace down to Q. Using the moduli interpretation, one cancheck the

Proposition 8.9. yK = −ε(E)yK in E(K)/E(K)tor, where ε(E) ∈ ±1 is the sign of thefunctional equation.

In particular, yK ∈ E(Q) + E(K)tor if and only if ε(E) = −1.30

Example 8.10. For the elliptic curve E = X0(32) : y2 = x3 + 4x, K = Q(√−7) satisfies

the Heegner hypothesis for X0(32). Then HK = K = Q(√−7) since Cl(K) = 0. Using

H∗ → E = Γ0(32)\H∗, we find

xK = yK =

(√−7− 1

2,

√−7 + 3

2

)∈ E(K).

Moreover, yK has infinite order, and ε(E) = +1.

Heegner used this construction to prove the following

Theorem 8.11. The elliptic curve E : y2 = x3 − n2x has ralg ≥ 1 when n is a prime ≡ 5(mod 8).

Example 8.12. For the elliptic curve E : y2 + y = x3 − x with conductor N = 37, K =Q(√−7) satisfies the Heegner hypothesis for X0(37). Then yK = (0, 0) ∈ E(K) has infinite

order, and this agrees with ε(E) = −1.

Next time we will relate the Heegner point yK with L(E, 1) via the Gross–Zagier formula,and prove that ran(E) = 1 =⇒ ralg(E) ≥ 1.

9. Lecture 9 (October 15, 2018)

Last time we introduced the Heegner hypothesis: an imaginary quadratic field K satisfiesthe Heegner hypothesis for X0(N) if

every p | N splits in K, (Heeg)

and used this to construct Heegner points xK ∈ X0(N)(HK) and yK ∈ E(K).

9.1. Gross–Zagier formula. Let EK be the base change of E from Q to K. Then L(EK , s)satisfies a functional equation

L(EK , s)↔ L(EK , 2− s)with sign of functional equation ε(EK). There is a simple formula for computing ε(EK).

Proposition 9.1. Assume (dK , N) = 1. Then ε(EK) = χK(−N), where χK : Z/|dK | →±1 is the quadratic character associated to K/Q.

Corollary 9.2. If K satisfies (Heeg), then ε(EK) = −1. More generally,

ε(EK) = −∏

p inert in K

(−1)ordpN .

To summarize, we have a diagram

ε(EK) = −1 +3 ran(EK) = odd

(Heeg)

4<

"*yK ∈ E(K)

?

==

so it is natural ask to ask if there is a relationship between yK ∈ E(K) and ran(EK) = odd.31

Theorem 9.3 (Gross–Zagier).

L′(EK , 1) =

∫E(C)

ω ∧ iω|dK |1/2|O×K//±1|2

〈yK , yK〉NT.

Here ω ∈ H0(EQ,Ω1) is such that ϕ∗ω = 2πifE(z) dz, where fE is the normalized newform

associated to E.

Remark. Since (∫E(C)

ω ∧ iω)· degϕ =

∫X0(C)

8π2f(z)f(z) dx dy = (f, f)

is the Petersson inner product, we can rewrite the Gross–Zagier formula as

L′(EK , 1) =(f, f)

|dK |12 |O×K/±1|2

· 〈yK , yK〉NT

degϕ.

Remark. The Heegner point yK depends on the choice of N ⊆ OK , but is determined up to

sign and torsion. It also depends on ϕ : X0(N)→ E, but〈yK , yK〉NT

degϕis canonical!

The proof of the Gross-Zagier formula is a computation. To understand it conceptually,we need to rephrase the formula.

Corollary 9.4. L′(EK , 1) = 0 ⇐⇒ 〈yK , yK〉NT = 0 ⇐⇒ yK is of infinite order. Thusran(EK) = 1 =⇒ ralg(EK) ≥ 1.

Remark. Comparison with the BSD formula gives

|X(EK)|12

?=

[E(K) : ZyK ]∏p cp(E) · |O×K |2 · c

where c is the Manin constant (conjectured to be 1): ϕ∗ωE = c · 2πifE(z) dz. In particular,this gives the conjectural implication

p - yKE(K)[p] = 0

?=⇒ X(EK)[p∞] = 0.

9.2. Back to E/Q. Now we explain how the result above descends to Q.

Definition 9.5. Let E(K)/Q be the quadratic twist of E by K, i.e., the unique elliptic curveover Q that is isomorphic to E over K but not isomorphic to E over Q. Explicitly, if E hasWeierstrass equation y2 = x3 + Ax+B, then

E(K) : dKy2 = x3 + Ax+B.

Remark. Consider ρE : GQ → Aut(VpE). Then ρE(K) = ρE ⊗ χK where χK : GQ GK/Q '±1.

Proposition 9.6. L(EK , s) = L(E, s)L(E(K), s).

Proof. Check by definition, or

L(EK , s) = L(IndGQ

GKρE, s) = L(ρE ⊕ ρE ⊗ χK , s) = L(ρE, s)L(ρE ⊗ χK , s).

Corollary 9.7. ran(EK) = ran(E) + ran(E(K)).32

By BSD, this predicts an identity on algebraic ranks, which is also easy to see directly.

Proposition 9.8. ralg(EK) = ralg(E) + ralg(E(K)).

Proof. Looking at the action of complex conjugation, we have

E(K)⊗Q ' E(Q)⊗Q⊕ E(K)(Q)⊗Q

into the eigenspaces for c = ±1.

Theorem 9.9. If ran(E) = 1, then ralg(E) ≥ 1.

Proof. By a theorem of Waldspurger (next time), one can choose K satisfying (Heeg) suchthat ran(E(K)) = 0. Then ran(EK) = 1 + 0 = 1, hence ralg(EK) ≥ 1. But ε(E) = −1, soycK = −ε(E)yK = yK in E(K)/E(K)tor. Therefore P = ycK + yK ∈ E(Q) is of infinite order,and ralg(E) ≥ 1.

Corollary 9.10. If ran(E) = 1, then

L′(E, 1)

Ω(E)R(E)∈ Q×.

Proof. Choose K as before. Then L′(EK , 1) = L′(E, 1)L(E(K), 1), whereL(E(K), 1)

Ω(E(K))∈ Q×.

The result follows from the Gross–Zagier formula using

∫E(C)

ω ∧ iω

|dK |12

Q×∼ Ω(E)Ω(E(K)).

Corollary 9.11. If ε(E) = −1 and ycK + yK ∈ E(Q) is torsion, then ran(E) ≥ 3.

Using this we can construct E/Q with ran(E) = 3.

Remark. There is no example of E/Q with provably correct ran(E) = 4.

Theorem 9.12 (Goldfeld, 1976). If there exists EQ with ran(E) ≥ 3, then the class numberof Q(

√−D) has an explicit lower bound

h(D) Cδ,E · (log |D|)1−δ

for all δ > 0, where Cδ,E is an effective constant.

Example 9.13 (Gauss elliptic curve). Consider E : y2 + y = x3 − 7x + 6 with N = 5077.Buhler–Gross–Zagier computed that ycK + yK is torsion, hence ran(E) ≥ 3. Together withGoldfeld, this solves Gauss’ class number problem.

9.3. Gross–Zagier formula for Shimura curves. Fix an imaginary quadratic field K/Q.For (N, dK) = 1, define N = N+N− where every p | N+ splits in K and every p | N− isinert in K.

Example 9.14. The Heegner hypothesis (Heeg) implies that N = N+.

Assume N− is square-free. Then ε(EK) = −(−1)#p|N−, so

ε(EK) =

+1 if #p | N− is odd,

−1 if #p | N− is even.

Definition 9.15. K satisfies the generalized Heegner hypothesis (Heeg∗) if N− is a square-free product of an even number of primes.

Next time we will construct yK for K satisfying this.33

10. Lecture 10 (October 17, 2018)

Last time we introduced the generalized Heegner hypothesis:N = N+N−, with N− a square-free product of an even number of primes,

p | N+ =⇒ p split in K,

p | N− =⇒ p inert in K.

(Heeg*)

Our goal is to construct Heegner points in this setting and fill in a similar diagram as lasttime

(Heeg*) +3

?

$,

ε(EK) = −1 +3 ran(EK) = odd

Heegner points yKuu

?55

10.1. Gross–Zagier formula for Shimura curves. Since #p | N− is even, we canconstruct the following

Definition 10.1. Let B = BN be the unique quaternion algebra over Q that is ramifiedexactly at p | N, i.e.,

Bv∼=

M2(Qv) if v - N−,division quaternion algebra over Qv if v | N−.

Example 10.2. Every quaternion algebra over Q is of the form B = Q1, i, j, ij wherei2 = a, j2 = b, ij = −ji.

• If a = 1, then we recover the matrix algebra B ' M2(Q) via i 7→ ( 1−1 ), j 7→ ( 1

b )and ij 7→ ( 1

−b ).

• If a is a prime p ≡ 1 (mod 4) and b is a prime q such that

(q

p

)6= 1, then B is

exactly ramified at p, q.

As an analogue of M0(N) =

(a bc d

): N | c

,

Definition 10.3. An Eichler order of level N+ is a subring O ⊆ OB (maximal order in B)of finite index, such that

Op =

M0(N+)p if p - N−,OBp if p | N−,

where OBp is the maximal order of Bp.

Now we can define the analogue of Γ0(N).

Definition 10.4. Γ(N+, N−) = γ ∈ O× : γγ = 1.

Example 10.5. If N− = 1, then Γ(N+, 1) = Γ0(N+).

Definition 10.6. Using Γ(N+, N−) → B×(R) ' GL2(R), the condition γγ = 1 impliesthat Γ(N+, N−) has image in SL2(R). We define the Shimura curve

X(N+, N−) := Γ(N+, N−)\H.34

Example 10.7.

• If N− = 1, then X(N+, N−) = Y0(N).• If N− 6= 1, then X(N+, N−) is already compact.

Remark. This is an example of a Shimura variety. One can rewrite X(N+, N−) adelically asfollows. Consider the compact open subgroup Kp = O×p ⊆ B×(Qp). Take Kf =

∏pKp ⊆

B×(Af ), again a compact open subgroup. Then Kf ∩B×1 (Q) = Γ(N+, N−) and

X(N+, N−) = Γ(N+, N−)\H∼= B×(Q)\B×(Af )×H±/Kf .

Notice H = SL2(R)/SO2(R).

Now we are ready to construct Heegner points using the moduli interpretation of Shimuracurves, similarly as in the case of modular curves:

• In the level 1 case,

X(1, N−)(C) = abelian surface A/C with OB → End(A)by sending τ ∈ H to C2/OB · ( τ1 ) via OB →M2(C).• In the general case,

X(N+, N−)(C) = A→ A′: cyclic OB-isogeny of degree (N+)2.

Example 10.8. If N− = 1, then Aτ = Eτ × Eτ .

Let K be an imaginary quadratic field satisfying (Heeg*), with Hilbert class field HK .

Definition 10.9. Define Heegner points xK = (A → A′) ∈ X(N+, N−)(HK) if A ∼ E2,A′ ∼ E ′2 and End(E) = End(E ′) = OK .

Definition 10.10. Using the modular parametrization ϕ : X(N+, N−)→ E, define

yK =∑

σ∈Gal(HK/K)

ϕ(σ(xK)) ∈ E(K).

Theorem 10.11 (Yuan–Zhang–Zhang). Assume K satisfies (Heeg∗). Then

L′(EK , 1) =(f, f)

|dK |12 |O×K/±1|2

· 〈yK , yK〉NT

degϕ.

Although this looks exactly the same as the original Gross–Zagier formula, the proof ismuch more complicated.

10.2. Waldspurger’s formula. To study the case ε(EK) = +1, we need a formula ofWaldspurger which was proved around the same time as Gross–Zagier and concerns thecentral value L(EK , 1).

Assume the hypothesis:

N− is a square-free product of an odd number of primes. (Wald)

This implies ε(EK) = +1. The goal is to construct “yK” such that yk ←→ L(EK , 1). But#p | N− is odd, so we cannot make a quaternion algebra B that is exactly ramified atp | N−. Instead, we consider the

35

Definition 10.12. Let B = BN−∞ be the unique quaternion algebra over Q ramified atp | N− ∪ ∞.

In particular, B(R) ' H (where i2 = j2 = −1, ij = −ji), so

B×1 (R) ∼= SU2 =

(x −yy x

)∈M2(C) : |x|2 + |y|2 = 1

is compact and does not act on H.

Definition 10.13. Let O ⊆ OB be an Eichler order of level N+. Define the Shimura set

X(N+, N−) := B×(Q)\B×(Af )/Kf

where Kf =∏

pO×p . (This is a 0-dimensional Shimura variety.)

Remark. A quaternion algebra B/Q is called indefinite if B(R) ' M2(R), and definite ifB(R) ' H.

Remark. Recall that for a number field F ,

F×\A×F,f/∏p

O×F,p∼→ Cl(OF ).

Similarly, for a definite quaternion algebra B,

X(1, N−) ∼= Cl(OB) = right ideal classes of OB.Next we introduce Heegner points on this 0-dimensional Shimura variety.

Definition 10.14. Fix an embedding K → B (with K ∩ O = OK). Then we have

Cl(OK) = K×\A×K,f/∏p

O×K,p → X(N+, N−) = Cl(O)

I 7→ IO.A point xK ∈ Cl(OK) (or its image) is called a CM point (or Gross point) on X(N+, N−).(Cl(OK) permutes these CM points.)

Definition 10.15. Let f ∈ Snew2 (N), which transfers under the Jacquet–Langlands corre-

spondence to a function ϕ : X(N+, N−)→ C with the same Hecke eigenvalues as f . (This isthe analogue of X(N+, N−)→ E in the indefinite case.) Define Waldspurger’s toric period

yK =∑

xK∈Cl(OK)

ϕ(xK)(|Aut(xK)|) ∈ C

where Aut(x) = γ ∈ B×(Q) : γx = x/±1, and

degϕ =∑

x∈X(N+,N−)

|ϕ(x)|2(|Aut(x)|).

Finally, we can state the

Theorem 10.16 (Waldspurger). Assume K satisfies (Wald). Then

L(EK , 1) =(f, f)

|dK |12 |O×K/±1|2

· |yK |2

degϕ.

Corollary 10.17. L(EK , 1) 6= 0 ⇐⇒ yK 6= 0.36

11. Lecture 11 (October 22, 2018)

Last time we stated Waldspurger’s formula: if K satisfies (Wald) for N = N+N−, then

L(EK , 1) =(f, f)

|dK |12 |O×K/±1|2

· |yK |2

degϕ

where yK is the toric period.

11.1. Examples of Waldspurger’s formula.

Example 11.1. Let E = X0(11), with Weierstrass equation y2 + y = x3 − x2 − 10x − 20and corresponding newform

f = fE = q∏n≥1

(1− qn)2(1− q11n)2 = η(z)2η(11z)2

= q − 2q2 − q3 + 2q4 + q5 + · · · .

Then K = Q(√−5) satisfies (Wald) (note 11 is inert in K), with N+ = 1 and N− = 11. Let

B = B11,∞ with maximal order O = OB, so that X(N+, N−) = X(1, 11) = Cl(OB).We make use of the following two formulas:

(1) (Eichler’s mass formula) If N− is prime, then∑

x∈Cl(OB)

1

|Aut(x)|=N− − 1

12.

(2)∏x

|Aut(x)| = denom

(N − 1

12

).

In our case, Eichler’s mass formula gives 1012

= 56

= 12

+ 13, so the second formula confirms

that |Cl(OB)| = 2 with |Aut(x1)| = 2 and |Aut(x2)| = 3.Consider the modular parametrization ϕ : X(N+, N−)→ C with Hecke eigenvalues equal

to those of f . It is easy to check that x1 7→ −1 and x2 7→ 1. Then yK = −2 + 3 = 1 (whichimplies ran(EK) = 0) and degϕ = 2 + 3 = 5. Now we compute that

L(EK , 1) = L(E, 1)L(E(K), 1)

= (0.25384 · · · )(0.65240 · · · ),(f, f) = 3.70309 · · · ,

=⇒ L(EK , 1)

(f, f)= 0.044721 · · · = 1

10√

5.

This matches up with1

|dK |12

· |yK |2

degϕ=

1√20· 1

5=

1

10√

5, as predicted by Waldspurger’s

formula.

Question. How does yK vary when K varies?

Answer. They fit into another modular form of weight 32.

Definition 11.2. Let Cl(O) = x1, · · · , xn (right ideal classes of O). For each i, define theleft order of xi

Ri = γ ∈ B : γxi = xi,37

andSi = γ ∈ Z + 2Ri : tr(γ) = 0,

which is a rank 3 Z-module together with a quadratic form given by the norm N.

Definition 11.3. Define a weight 32θ-series by

gi :=1

2

∑v∈Si

qNv.

Thus the D-th Fourier coefficient of gi encodes the maps xQ(√−D) ∈ CM points 7→ xi ∈

X(N+, N−) under Cl(OK)→ Cl(O).

Definition 11.4. Associated to f , define

g :=n∑i=1

ϕ(xi)gi =∑n≥1

bnqn.

Theorem 11.5. yK = b|dK |.

Example 11.6. Let E = X0(11) and Cl(OB) = x1, x2 as before. Then we can computethat

g1 =1

2

∑x,y,z∈Z

x≡y (mod 2)

qx2+11y2+11z2 =

1

2+ q4 + q11 + 2q12 + q16 + 2q20 + · · · ,

g2 =1

2

∑x,y,z∈Z

x≡y (mod 3)y≡z (mod 2)

qx2+11y2+33z2

3 =1

2+ q3 + q12 + 3q15 + 3q16 + 3q20 + · · · ,

and hence g = −g1 + g2 = q3 − q4 − q11 + 2q16 + q20 + · · · . In particular, yK = 1 forK = Q(

√−5)! Indeed, for E = X0(11),

BSD formula for EK ⇐⇒ |X(E/K)| ?= |b|dK ||2.

In particular, X(E/Q(√−5)) = 0!

The first two non-trivial examples are given by

g = q3 − q4 − q11 + 2q16 + q20 + · · ·+ 3q67 + · · ·+ 4q91 + · · · .Indeed, X(E/Q(

√−67)) = (Z/3)2 and X(E/Q(

√−91)) = (Z/2)4.

Remark. By studying Fourier coefficients of weight 32

forms, one can show there are a lot of

K such that yK 6= 0, which implies that ran(E(K)) = 0 for many K. Recall that this is theinput for deducing BSD over Q from BSD over K.

11.2. Back to Kolyvagin. Recall that we have constructed Heegner points on modu-lar curves, Shimura curves or Shimura sets. Our next goal is to construct classes c(`) ∈H1(K,E[p]) such that loc`(c(`)) is ramified, i.e., its image in H1

sing(K`, E[p]) is nonzero.As a naive try, say K satisfies (Heeg*). Then we have Heegner points yK ∈ E(K). Using

the connecting homomorphism

δ :E(K)

pE(K)→ H1(K,E[p]),

38

we obtain δ(yK) ∈ H1(K,E[p]).Problem: If ` /∈ S = x | Np∞, then δ(yK) is unramified at `.

The idea is to find another elliptic curve E(`) such that E[p]∼→ E(`)[p] and E(`) has

conductor N`. If so, choose K satisfying (Heeg*) for N`. This gives yK(`) ∈ E(`)(K) andc(`) := δ(yK(`)) ∈ H1(K,E(`)[p]) ∼= H1(K,E[p]). Then it is possible that c(`) is ramified at`, as ` | N`.

To summarize, we can produce ramification by raising the level under congruences mod p.

Question. Given f ∈ Snew2 (N), when can we find g ∈ Snew

2 (N`) such that f ≡ g (mod p)(i.e., aq(f) ≡ qq(g) (mod p) for almost all primes q)?

In general this is not possible. For example, when the level is small, the number of mod-ular forms is limited. We can reformulate this question in terms in Galois representations.Consider ρf : GQ → GL2(Zp) and the residual representation ρf : GQ → GL2(Fp).

Question. When can we find g ∈ Snew2 (N`) such that ρf ∼= ρg?

Under this reformulation, we immediately see a necessary condition. Since ρf |GQ`

∼= ρg|GQ`

and ρf is unramified at `, we want ρg to be unramified (even though ρg is ramified at `),

so g has multiplicative reduction at `. By Tate’s uniformization, Frob` is sent to ±(` ∗0 1

).

Taking tr(Frob`), we get a`(f) ≡ ±(`+ 1) (mod p).

Definition 11.7. ` - Np is called a level-raising prime for f (mod p) if

a`(f) ≡ ±(`+ 1) (mod p).

Ribet shows this is almost a sufficient condition:

Theorem 11.8 (Ribet, 1990). If ρf is irreducible, and ` is a level-raising prime for f(mod p), then there exists g ∈ S`-new

2 (N`) such that ρf ' ρg.

Example 11.9. Let E = X0(11) and p = 2. It is clear that

a`(f) ≡ ±(`+ 1) (mod 2) ⇐⇒ a`(f) is even,

so ` = 7 is level-raising. The corresponding g ∈ Snew2 (77) is the newform with label 77a:

2 3 5 7 11 13 17 19f = fE −2 −1 1 −2 1 4 −2 0g = 77a 0 −3 −1 −1 −1 −4 2 −6

12. Lecture 12 (October 24, 2018)

Last time we stated the

Theorem 12.1 (Ribet). Let f ∈ S2(N) such that ρf is irreducible, and ` be a level-raisingprime for f (mod p), i.e., a`(f) ≡ ±(`+ 1) (mod p). Then there exists g ∈ S`-new

2 (N`) suchthat ρf ' ρg.

39

12.1. Geometric interpretation of Ribet’s theorem: Ihara’s lemma. Recall that themodular curve X0(N)/Q has the moduli interpretation

(E,C) : E elliptic curve, C ⊆ E[N ] cyclic subgroup of order N.

There is a correspondence

X0(N`)π1

yy

π2

%%

X0(N) X0(N),

where

π1 : (E,C) 7→ (E,C[N ]), π2 : (E,C) 7→ (E/C[`], C/C[`])

which make sense since C ' Z/N`.

Example 12.2. When N = 1, X0(`) parametrizes isogenies (E → E ′) of degree `, and

π1 : (E → E ′) 7→ E, π2 : (E → E ′) 7→ E ′.

The Hecke operator can be defined as

T` := π2∗π∗1 = π1∗π

∗2 ∈ End(J0(N)).

Consider the map π∗1 + π∗2, which fits into the diagram

H0(X0(N),Ω1)⊕2π∗1+π∗2

// H0(X0(N`),Ω1)

S2(N)⊕2 // S2(N`).

Concretely, π∗1(f(z)) = f(z) (via z 7→ z) and π∗2(f(z)) = `f(`z) (via z 7→ `z). These are`-old forms, so

S`-new2 (N`) = coker(π∗1 + π∗2).

To deal with mod p congruences (which require Zp-structures), we look at

π∗1 + π∗2 : H1et(X0(N),Zp)

⊕2 → H1et(X0(N`),Zp).

After inverting p, this is injective and the Hecke eigenvalues on coker(π∗1 + π∗2)Qp are exactlygiven by those on S`-new

2 (N`).

Definition 12.3. Let T be the Hecke algebra generated by Tq for all q - N`. Let χf : T→ Zp

be the Hecke eigenvalues associated to f . Let m = mf := ker(Tχf→ Zp Fp) be the maximal

ideal of T encoding congruences with f .

To prove Ribet’s theorem, it suffices to prove that the map

π∗1 + π∗2 : H1et(X0(N),Zp)

⊕2m → H1

et(X0(N`),Zp)m

(localization at m = mf ) has cokernel containing a point in characteristic 0, i.e., coker(π∗1+π∗2)is not torsion.

40

Theorem 12.4 (Ihara’s lemma). Assume ρf is irreducible. Then

π∗1 + π∗2 : H1et(X0(N),Fp)

⊕2m → H1

et(X0(N`),Fp)m

is injective.

Proof that Ihara implies Ribet. If coker(π∗1 +π∗2)Zp is torsion, then Ihara’s lemma implies thatcoker(π∗1 + π∗2)Zp is trivial, so (π∗1 + π∗2)Zp is an isomorphism. To get a contradiction, look atthe dual map

(π1∗, π2∗) : H1et(X0(N`),Zp)m → H1

et(X0(N),Zp)⊕2m

which is also an isomorphism by duality. The composition

H1et(X0(N),Zp)

⊕2m

∼→ H1et(X0(N),Zp)

⊕2m

is an isomorphism, with 2× 2 matrix given by(`+ 1 T`T` `+ 1

)m

≡(

`+ 1 ±(`+ 1)±(`+ 1) `+ 1

)(mod p).

This is a contradiction to the composition being an isomorphism.

12.2. Geometry of modular curves in characteristic p. Recall that X0(N) has moduliinterpretation (E,C) with C ⊆ E[N ]. The group E[N ] still behaves well in characteristicp - N (i.e., is etale). In particular, if p - N , then

E[N ](Fp) ' (Z/N)2.

But E[p] behaves differently in characteristic p:

E[p](Fp) '

Z/p if E is ordinary,

0 if E is supersingular.

Equivalently,

End(E/Fp) '

O ⊆ OK if E is ordinary,

O ⊆ Op,∞ if E is supersingular,

where K is an imaginary quadratic field, and Op,∞ is the maximal order in Bp,∞.To define an integral model of X0(N), we need to change the moduli interpretation.

Theorem 12.5 (Igusa, Deligne–Rapoport, Katz–Mazur). X0(N) has a model over Z. It isflat, projective and of relative dimension 1 over Z.

(1) If p - N , then X0(N)Zp has the same moduli interpretation as before; X0(N)Fp issmooth, and has Newton stratum

X0(N)Fp = X0(N)ordFp ∪X0(N)ss

Fp

according to whether E is ordinary or supersingular, where the supersingular locusX0(N)ss

Fpis a finite set of points.

(2) If p ‖ N , then X0(N)Zp has moduli interpretation

S/Zp 7→

(E,C) :E/S elliptic curve, C ⊆ E[p] finite flatgroup scheme over Zp of rank N

;

X0(N)Fp is no longer smooth but is semistable, with two irreducible components iso-morphic to X0(N/p)Fp and intersecting along X0(N/p)ss

Fp= X0(N)ss.

41

(3) If p2 | N , then X0(N)Zp has moduli interpretation in terms of Drinfeld level structure:

C =⊕a∈Z/p

[aP ], P ∈ E[p]

as a Cartier divisor on E; X0(N)Fp is not smooth and not semistable.

Remark. If p ‖ N , the picture looks like

Let us describe the gluing datum. Over Fp, we have

X0(N/p)α1

%%

X0(N/p)α2

yy

X0(N)π1

yy

π2

%%

X0(N/p) X0(N/p)

where (assuming N/p = 1 for simplicity)

π1 : (Edeg p→ E ′) 7→ E, π2 : (E

deg p→ E ′) 7→ E ′,

α1 : E 7→ (EFrobp→ E(p)), α2 : E 7→ (E(p) Verp→ E).

Notice

π1 α1 = Id, π2 α2 = Id,

π1 α2 = Frobp, π2 α1 = Verp .

The two copies of X0(N/p) are glued via E ↔ E(p), as (E(p))(p) ' E if E is supersingular.

Next time we will describe X0(N)ssFp

explicitly and prove Ihara’s lemma.

13. Lecture 13 (October 29, 2018)

Last time we saw that the special fiber X0(N)Fp is smooth if p - N . Our goal is to describethe supersingular locus X0(N)ss

Fp, which is a finite set.

13.1. The supersingular locus.

Proposition 13.1. X0(N)ssFp' X(N, p), the Shimura set associated to Bp,∞, which is in

turn isomorphic to Cl(O) for the Eichler order O of level N in Bp,∞.

Remark. This is an instance of “switching invariants”: X0(N)ssFp

is indefinite at ∞ and split

at p, while X(N, p) is definite at ∞ and ramified at p.42

Idea of proof. Given (E,C) ∈ X0(N)ssFp

, we have End(E) ' Op,∞ (maximal order in Bp,∞),

under which the subring End(E,C) corresponds to O (Eichler order of level N).Given any point (E ′, C ′) ∈ X0(N)ss

Fp, look at I = Hom((E,C), (E ′, C ′)), which is a rank 4

Z-module and admits a right action of O = End(E,C): for f ∈ I and g ∈ O, f · g := f g.This construction gives a map

X0(N)ssFp → X(N, p) = Cl(O)

(E ′, C ′) 7→ I = Hom((E,C), (E ′, C ′)).

One can check this is a bijection.

Last time we saw that Ihara’s lemma implies Ribet’s theorem for level-raising congruences.Recall

Theorem 13.2 (Ihara’s lemma). Assume ρf is irreducible (so that the maximal ideal m =mf ⊆ T is “non-Eisenstein”). Then

H1et(X0(N),Fp)

⊕2m → H1

et(X0(N`),Fp)m

is injective.

Idea of proof. Use p-adic comparison theorem between H1et(X0(N),Fp) and H1

dR(X0(N)/Fp)and reduce to showing that

π∗1 + π∗2 : H0(X0(N)Fp ,Ω1)⊕2 → H0(X0(N`)Fp ,Ω

1)

is injective.Suppose (ω1, ω2) ∈ ker(π∗1 + π∗2). By looking at the Hecke action of T`, we see that if

(E ′, C ′) ∈ T`(E,C), then

(E ′, C ′) ∈ div0(ωi) ⇐⇒ (E,C) ∈ div0(ωi)

for i = 1, 2. Thus div0(ωi) is supported on the whole X0(N)ssFp

(if (E,C) ∈ X0(N)ordFp

, then

T n` (E,C) : n ≥ 1 is infinite).But there is an explicit differential form (Hasse invariant) A ∈ H0(X0(N)Fp , ω

p−1) ofweight p− 1 with simple zeros at exactly X0(N)ss

Fp. If p− 1 > 2, this is a contradiction!

If p = 3 (resp. p = 2), then ωi = c ·A (resp. ωi = c ·A2). Both cases are impossible, sincethe Hecke action on A is given by an Eisenstein ideal: Tq ≡ q + 1 (mod p).

13.2. Geometry of Shimura curves. We want to carry over the geometry of modularcurves to the case of Shimura curves. Recall the Shimura curve X(N+, N−) has moduliinterpretation (A, ι, C) where:

• A is an abelian surface,• ι : ON− → End(A),• C ⊆ A[N ] of order (N+)2 and cyclic as ON−-module.

Analogous to X0(N) = X(N, 1), we can extend this moduli interpretation to characteristicp - N = N+N−. But at p | N , changes need to be made (especially more difficult whenp | N−)!

Let us begin with the classification of abelian surfaces with special endomorphisms incharacteristic p. Let A/Fp be an abelian surface with ON−-action. Then A is either:

• ordinary : A[p] ' (Z/p)2 ⇐⇒ A is isogenous to E2 for an ordinary elliptic curve E.43

• supersingular : A[p] = 0 ⇐⇒ A is isogenous to E2 for a supersingular E.

If A is supersingular, we say that A is superspecial if A is isomorphic to E2 for a supersingularE. Similarly to the case of elliptic curves, we have

End(A) =

O ⊆M2(OK) if A is ordinary,

O ⊆M2(Op,∞) if A is supersingular,

where K is an imaginary quadratic field.

Remark. By definition, every superspecial A is supersingular. When p - N−, the converse istrue: A is superspecial if and only if A is supesingular.

The geometry of Shimura curves is summarized in the

Theorem 13.3 (Buzzard, Drinfeld). Assume p2 - N . Then X(N+, N−) has a model overZp which is flat and projective.

(1) If p - N , then X(N+, N−)Fp is smooth, with a distinguished finite set of markedpoints X(N+, N−)ss

Fp. Moreover,

X(N+, N−)ssFp ' X(N+, pN−),

which is a definite Shimura set (note that pN− has an odd number of prime factors).

Remark. In this case, X(N+, N−)ssFp

= X(N+, N−)s.spFp

.

(2) If p ‖ N+, then X(N+, N−)Fp is not smooth but is semistable, with two irreduciblecomponents each isomorphic to X(N+/p,N−) and intersecting along

X(N+, N−)ssFp = X(N+/p,N−)ss

Fp

(1)= X(N+/p, pN−).

Remark. Cases (1) and (2) are analogous to the situation for X0(N).

(3) If p | N−, then X(N+, N−)Fp is not smooth but is semistable. Its geometry is de-scribed by Ceredink–Drinfeld p-adic uniformization:

X(N+, N−)Fp = X1 ∪X2,

where each Xi is a family of P1’s indexed by the Shimura set X(N+, N−/p) (switchinginvariants). Each P1 in X1 intersects with exactly p+ 1 P1’s in X2 (and vice versa),forming P1(Fp). Finally,

X1 ∩X2 = (p+ 1) · |X(N+, N−/p)| = X(pN+, N−/p).44

The geometry of X(N+, N−) is summarized in the picture:

• For p - N , X(N+, N−)ssFp

= X(N+, pN−).

• For p | N−, the irreducible components of X(N+, N−)Fp are X(N+, N−/p).

14. Lecture 14 (October 31, 2018)

14.1. Inertia action on cohomology. The motivation is to understand

H1sing(K`, E[p]) = H1(I`, E[p])Fr`=1,

where I` is the inertia subgroup at `. Thus we need to understand the action of I` onE[p] = H1

et(E,Fp)(1).In general, let us switch notation to the local situation. Suppose X is a smooth projective

variety over Qp, and K = Qurp is the completion of the maximal unramified extension of Qp

45

(so Ip = Gal(K/K)). Let O = OK(

= Zurp

)and k = O/p = Fp. Let Λ = Z/` or Q` be the

coefficient ring.Recall the wild inertia subgroup P ⊆ I is defined as P = Gal(K/Kt) where Kt =⋃p-nK(p1/n) is the maximal tamely ramified extension of K. Then I/P '

∏`6=p Z`(1).

Qp K

I

Kt

P

K

Definition 14.1. Define the tame `-adic character to be the map t : I → Z`(1) given by

σ 7→σ(p1/`m)

p1/`m

∈ µ`mm≥1 = Z`(1).

Theorem 14.2 (Grothendieck). Let V = H i(et)(XK ,Λ) with Λ = Q`, which has an action

of I = Gal(K/K). Then there exists an open subgroup J ⊆ I and a nilpotent matrixN : V (1) → V such that for all σ ∈ J , the action of σ on V is given by exp(t(σ)N) (henceunipotent).

Definition 14.3. The operator N is called the monodromy operator.

Remark. The geometric analogue is

Z`(1)ff

&&

I = Gal(K/K)oo

OO

Spec k

// SpecO SpecK? _oo

π1(D − 0) ' Zz = 0point

D = |z| < 1disk

D − 0punctured disk

After going around a loop in SpecK, we get an automorphism of H i(XK ,Λ) (monodromyin geometry).

Example 14.4. Let X/Qp be an elliptic curve with multiplicative reduction. Then by Tate’suniformization theorem, the action of σ ∈ I on V = H1(XK ,Λ) (or T`X) is given by(

1 t(σ)1

)= exp

(0 t(σ)

0

),

so N =

(0 10 0

).

14.2. Nearby cycle sheaves and monodromy filtration. Our goal is to describeH i(XK ,Λ)and N using the geometry of Xk = XFp

. Let XO be a proper integral model of XK .

Xk i

//

XO

XK? _

joo

Spec k

// SpecO SpecK? _oo

Definition 14.5. The nearby cycle sheaf is

RΨΛ := i∗Rj∗Λ ∈ D(Xk).46

Then

H i(XK ,Λ) = H i(XO, Rj∗Λ) (for any map j)

= H i(Xk, i∗Rj∗Λ) (proper base change)

= H i(Xk, RΨΛ),

so the slogan is

cohomology of generic fiber = cohomology of special fiber for nearby cycle sheaf.

Thus the nearby cycle sheaf sees the cohomology of “nearby” fibers.

Example 14.6. If XO → SpecO is smooth, then RΨΛ ∼= Λ and H i(XK ,Λ) ' H i(Xk,Λ).In this case, N = 0 (unramified).

In general, there is an action

N : RΨΛ(1)→ RΨΛ

which is nilpotent. More precisely, Nn+1 = 0 for n = dimX.

Definition 14.7.

(1) The kernel filtration is given by Fi := kerN i+1:

0 ⊆ F0 ⊆ F1 ⊆ · · · ⊆ Fn = RΨΛ.

(2) The image filtration is given by Gj := imN j:

RΨΛ ⊇ G1 ⊇ G2 ⊇ · · · ⊇ Gn ⊇ 0.

(3) The monodromy filtration is given by Mr :=∑

i−j=r Fi ∩Gj:

0 ⊆M−n ⊆ · · · ⊆Mn ⊆ RΨΛ.

The monodromy filtration is characterized by the properties:

(1) N(Mr(1)) ⊆Mr−2.

(2) N r : grMr RΨΛ(r)∼→ grM−rRΨΛ is an isomorphism, where grMr = Mr/Mr−1.

14.3. Rapoport–Zink weight spectral sequence.

Definition 14.8. XO is semistable if

(1) XO is regular and flat over O.(2) Xk is a divisor of XO with normal crossings, i.e., its singularities are given byO[x1, · · · , xn]/(x1 · · ·xs − p) for some s ≤ n.

We further say that XO is strictly semistable if

(3) all irreducible components of Xk are smooth (“simple” normal crossings).

For example, the nodal curve is semistable, but not strictly semistable. The union of twocurves intersecting transversally is strictly semistable.

Definition 14.9. Assume XO is strictly semistable. Then Xk = X1 ∪ · · · ∪Xm where eachXi is a smooth irreducible component (of codimension 1 in XO). Define for any subsetJ ⊆ 1, · · · ,m,

XJ :=⋂i∈J

Xi.

47

For any p ≥ 0, let

X(p) :=∐

J⊆1,··· ,m|J |=p+1

XJ ,

which is a smooth projective variety over k of codimension p in Xk.

For any p ≥ 0, we have

• pullback: Hq(X(p),Λ)→ Hq(X(p+1),Λ),• pushforward: Hq(X(p),Λ)(−1)→ Hq+2(X(p−1),Λ).

Theorem 14.10 (Rapoport–Zink, Saito). Assume XO is strictly semistable. Then we havea spectral sequence

Ep,q1 = Hp+q(Xk, grM−pRΨΛ)⇒ Hp+q(XK ,Λ),

which degenerates at E2 if Λ = Q`. Moreover,

Ep,q1 =

⊕i≥0i≥−p

Hq−2i(X(p+2i),Λ)(−i)

where the differentials on E1 are sums of pushforwards and pullbacks, and N : Ep,q1 →

Ep+2,q−21 is given by ⊗t.

Remark. The terminology of “weight” spectral sequence comes from the

Conjecture 14.11 (Weight monodromy). Let Λ = Q`. Then grM−rHi(XK ,Λ) is pure of

weight i+ r (i.e., the action of Frp has eigenvalues of absolute value pi+r2 ).

15. Lecture 15 (November 7, 2018)

16. Lecture 16 (November 12, 2018)

17. Lecture 17 (November 14, 2018)

18. Lecture 18 (November 19, 2018)

18.0. Rankin–Selberg L-functions for GLn × GLn−1. As a motivation, recall that theway we proved the rank part of BSD is by going through

L(E, s) = L(f, s).

• The left-hand side is a special case of a “motivic L-function” L(ρE, s), where ρE isthe motive associated to E.• The right-hand side is an “automorphic L-function” L(π, s), where π is an automor-

phic representation on GL2(A) associated to f ∈ π.

The global Langlands correspondence states that every motivic L-function arises as an auto-morphic L-function. In this way, arithmetic questions about motives (e.g. counting points onalgebraic varieties) can be studied using analytic properties of automorphic representations,which are often easier.

Our goal is to study generalizations to the arithmetic of L(s, π×π′), where π (resp. π′) isa cuspidal automorphic representation on GLn(A) (resp. GLn−1(A)), for A = AK the ringof adeles of a number field K/Q.

48

Example 18.1. Let n = 2. Taking π to be the automorphic representation correspondingto f and π′ to be the trivial representation gives L(f, s).

Today we will define L(s, π × π′) and state its basic properties. Similarly to the case ofthe Riemann zeta function, the three basic properties are analytic continuation, functionalequation, and Euler product (which relates to the problem of counting points on varieties).The study follows the template of Tate’s thesis.

18.1. Global zeta integrals.

Definition 18.2 (Global zeta integrals). Let f ∈ π and ϕ ∈ π′. Define

Z(f, ϕ, s) :=

∫[GLn−1]

f

(g

1

)ϕ(g)| det g|s−

12 dg,

where [GLn−1] = GLn−1(K)\GLn−1(A).

Example 18.3. Let n = 2, and f ∈ Snew2 (N). Recall the L-function

L(f, s) =∑n≥1

anns

=∏p

Lp(f, s)

and the completed L-function

Λ(f, s) = (2π)−sΓ(s)L(f, s) =

∫ ∞0

f(it)tsdt

t.

Let us rewrite this in adelic language. Every point x + iy ∈ H corresponds to

(y x

1

)∈

GL2(R), and f corresponds to f

(y

1

)= f(iy)y. Then Λ(f, s) generalizes to∫

[GL1]

f

(y

1

)|y|s−1 dy,

so that L(f, s) = Z(f,1, s+ 12).

The basic properties of the global zeta integrals are summarized in the

Theorem 18.4 (Properties of global zeta integrals).

(1) Z(f, ϕ, s) converges to an analytic function on s ∈ C.(2) Z(f, ϕ, s) satisfies a functional equation

Z(f, ϕ, s) = Z(f , ϕ, 1− s),

where f(g) = f(tg−1) and similarly for ϕ.(3) Suppose f =

⊗fv ∈ π =

⊗v πv and ϕ =

⊗ϕv ∈ π′ =

⊗v π′v. Then Z(f, ϕ, s) has

an Euler product

Z(f, ϕ, s) =∏v

Zv(fv, ϕv, s)

where Zv is to be defined.

Sketch of proof.

(1) This is clear, since f is a cusp form and hence rapidly-decreasing along cusps.49

(2) Use the automorphism g 7→ tg−1.(3) Euler product uses the uniqueness of Whittaker models, which will be explained in

the following.

Before defining the Whittaker models, we need to introduce the Whittaker(–Fourier) ex-pansion. Fix an additive character ψ : K\A→ C×.

Definition 18.5. For f ∈ π, define

Wf (g) =

∫[Nn]

f(ng)ψ−1(n) dn,

where Nn is the maximal unipotent subgroup of GLn consisting of all the upper-triangularunipotent matrices, and

ψ

1 x12 ∗

1 x23

. . . xn−1,x

1

= ψ(x12 + x23 + · · ·+ xn−1,n).

Theorem 18.6 (Jacquet, Piatetski-Shapiro, Shalika).

f(g) =∑

γ∈Nn−1(K)\GLn−1(K)

Wf

((γ

1

)g

).

Example 18.7. For n = 2,

Wf (g) =

∫[Ga]

f

((1 x

1

)g

)ψ−1(x) dx

is the familiar Fourier coefficient, and

f(g) =∑γ∈K×

Wf

((γ

1

)g

)=∑γ∈K×

Wf,ψγ (g)

where ψγ(x) = ψ(γx). Here Wf,ψγ (g) is the generalization of an · qn.

Remark. If n > 2, difficulty for the theorem arises because Nn−1 is not abelian. One has tocarry out a subtle induction on n.

Now we define the Whittaker model.

Definition 18.8. Let

IndGLnNn

(ψ) = W ∈ C∞(GLn(A)) : W (ng) = ψ(n)W (g), n ∈ Nn(A), g ∈ GLn(A) ,called the space of (global) Whittaker functions.

By definition, Wf (g) ∈ IndGLnNn

(ψ).

Definition 18.9. The (global) Whittaker model is

W (π, ψ) = Wf (g) : f ∈ π.50

We have a GLn(A)-equivariant map

π → W (π, ψ)

f 7→ Wf .

This provides a way of understanding automorphic representations in terms of concretefunctions.

Now we can use Whittaker functions to prove the Euler product; this technique is called“unfolding”.

Z(f, ϕ, s) =

∫[GLn−1]

f

(g

1

)ϕ(g)| det g|s−

12 dg

=

∫[GLn−1]

∑γ∈Nn−1(K)\GLn−1(K)

Wf

((γ

1

)(g

1

))ϕ(g)| det g|s−12 dg

=

∫Nn−1(K)\GLn−1(A)

Wf

(g

1

)ϕ(g)| det g|s−

12 dg

=

∫Nn−1(A)\GLn−1(A)

Wf

(g

1

)Wϕ(g)| det g|s−

12 dg,

where the last step is by the property of Whittaker functions. Note that Nn−1(A)\GLn−1(A)is factorizable!

Analogous to the global situation, we make the following definitions.

Definition 18.10. Let

IndGLnNn

(ψv) = W ∈ C∞(GLn(Kv)) : W (ng) = ψv(n)W (g), n ∈ Nn(Kv), g ∈ GLn(Kv)be the space of local Whittaker functions.

Definition 18.11. A local Whittaker model is a nontrivial GLn(Kv)-equivariant map

πv → IndGLnNn

(ψv),

i.e., a nonzero element in HomGLn(Kv)(πv, IndGLnNn

(ψv)).

Theorem 18.12 (Uniqueness of local Whittaker model: Gelfand–Kazhdan). Let πv be asmooth irreducible admissible representation of GLn(Kv). Fix a nontrivial additive characterψv : Kv → C×. Then

dim HomGLn(Kv)(πv, IndGLnNn

(ψv)) ≤ 1.

The proof uses Bessel distributions.As a consequence, if f =

⊗fv is factorizable, then

Wf (g) =∏v

Wfv(gv).

Definition 18.13 (Local zeta integrals).

Zv(fv, ϕv, s) :=

∫Nn−1(Kv)\GLn−1(Kv)

Wfv

(gv

1

)Wϕv(gv)| det gv|s−

12 dgv.

51

Corollary 18.14.

Z(f, ϕ, s) =∏v

Zv(fv, ϕv, s).

18.2. Relation between Z(f, ϕ, s) and L(s, π × π′).

Theorem 18.15 (Properties of local zeta integrals).

(1) Zv(fv, ϕv, s) converges for Re s 0.(2) Zv(fv, ϕv, s) ∈ C(q−s) and hence has meromorphic continuation to C.(3) Zv(fv, ϕv, s) : fv ∈ πv, ϕv ∈ π′v is a C[q±s]-ideal in C(q−s) and hence has a gener-

ator Pπv ,π′v(q−s)−1, where P ∈ C[X] and P (0) = 1.

Definition 18.16. Define the local L-factor to be

L(πv × π′v, s) := Pπv ,π′v(q−s)−1.

Theorem 18.17 (Local functional equation).

Zv(fv, ϕv, s) = ωπ′(−1)n−1γ(s, πv × π′v, ψv) · Zv(fv, ϕv, 1− s),

where γ(s, πv × π′v, ψv) is defined using the uniqueness of local Whittaker models.

Definition 18.18.

ε(s, πv × π′v, ψv) := γ(s, πv × π′v, ψv) ·L(s, πv × π′v)

L(1− s, πv × π′v).

18.3. Global L-functions.

Definition 18.19.

L(s, π × π′) =∏v-∞

L(s, πv × π′v),

Λ(s, π × π′) =∏v

L(s, πv × π′v),

ε(s, π × π′) =∏v

ε(s, πv × π′v, ψv).

By the product formula, ε(s, π × π′) is independent of the choice of ψ.

Theorem 18.20.

(1) L(s, π × π′) has analytic continuation to C.(2) There is a functional equation

Λ(s, π × π′) = ε(s, π × π′)Λ(s, π × π′).

The proof uses Z(f, ϕ, s) and Zv(fv, ϕv, s).52

19. Lecture 19 (November 26, 2018)

19.1. General Waldspurger formula. Recall the Waldspurger hypothesis: let E/Q bean elliptic curve, and K/Q be an imaginary quadratic field satisfying

#p | N : p inert in K is odd, i.e., #p | N− is odd. (Wald)

Thus (Wald) implies that ε(EK) = +1. The Waldspurger formula states that

L(EK , 1)

(f, f)=

1

|dK |1/2· |yK |

2

degϕ,

where yK is a toric period on a certain definite quaternion algebra and ϕ is the modularparametrization of E. As a consequence, L(EK , 1) 6= 0 if and only if yK 6= 0. This is crucialin proving ran(E) = 0 =⇒ ralg(E) = 0.

This formula is actually much more general. Notice that under the association

elliptic curve E newform f ∈ Snew2 (N)

automorphic representation π on GL2(A)(with central character),

we have

L(EK , s) ∼ L(πK , s) = L(πK ⊗ 1K , s)

where 1K : K×\A× → C× is the trivial character, and ∼ means the L-functions differ by ashift.

In general, take a number field F with ring of adeles A = AF , a quadratic extension K/F ,a quaternion algebra B/F (including M2(F )), a cuspidal automorphic representation π onG := B×/F×, and a character χ on H := K×/F×. Then an embedding K → B induces anembedding H → G (as algebraic groups over F ).

Now we are ready to generalize the toric period yK :

Definition 19.1. The automorphic H-period is defined to be

℘H(f) :=

∫[H]

f(h)χ(h) dh ∈ C

for f ∈ π, where [H] = H(F )\H(AF ).

Notice ℘H : f 7→ ℘H(f) defines a linear functional

℘H ∈ HomH(A)(π ⊗ χ,C) =∏′

v

HomH(Fv)(πv ⊗ χv,C).

Theorem 19.2 (Tunnell, Saito).

(1) (Multiplicity one)

dim HomH(Fv)(πv ⊗ χv,C) ≤ 1.

(2) (ε-dichotomy) Let G0(Fv) = PGL2(Fv) and G1(Fv) = B×v /F×v . Suppose π0

v is an irre-ducible representation of G0(Fv), and π1

v = JL(π0v) is its Jacquet–Langlands transfer

to G1(Fv) (which is nonzero only when π0v is discrete series). Then

dim HomH(Fv)(π0v ⊗ χv,C) + dim HomH(Fv)(π

1v ⊗ χv,C) = 1.

53

Moreover, HomH(Fv)(π0v ⊗ χv,C) 6= 0 if and only if the local root number

ε

(1

2, π0

v ⊗ χv)

= χv(−1)ηv(−1),

where η = ηK/F .

Example 19.3. Let π be an automorphic representation coming from an elliptic curve E/Q,and χ = 1K . Then

ε

(1

2, πv ⊗ χv

)= εv(E/K).

There are two cases:

• If v - N , then εv(E/K) = +1 and π1v = 0. These imply dim HomH(Fv)(π

0v⊗χv,C) = 1.

• If v | N , then

εv(E/K) =

+1 if v splits in K,

(−1)ordv N if v is inert in K.

Moreover, χv(−1) = +1 and ηv(−1) = +1 (as v - dK). So dim HomH(Fv)(π0v⊗χv,C) =

1 when v | N+, and dim HomH(Fv)(π1v ⊗ χv,C) = 1 when v | N−.

Thus (Wald) impliesdim HomH(A)(π ⊗ χ,C) = 1.

This is the representation-theoretic interpretation of the Waldspurger condition.

Theorem 19.4 (Waldspurger’s nonvanishing criterion). The following are equivalent:

(1) ℘H 6= 0.(2) HomH(A)(π ⊗ χ) 6= 0 and L(π ⊗ χ, 1

2) 6= 0.

This is enough for the rank part of BSD. Indeed, Waldspurger also proves a refined formulafor L(π ⊗ χ, 1

2) in terms of the period ℘H . One constructs a canonical element

αv ∈ HomH(Fv)(πv ⊗ χv,C)⊗ HomH(Fv)(πv ⊗ χv,C)

using matrix coefficients: for f ∈ πv and f ∈ πv,

αv(f, f) :=

∫H(Fv)

〈πv(h)f, f〉χv(f) dh.

Remark. If πv is unramified and f, f are spherical such that 〈f, f〉 = 1, then

αv(f, f) =ζFv(2)L(1

2, πv ⊗ χv)

L(1, ηv)L(1, πv,Ad)=: Lv

(1

2, π ⊗ χ

).

Theorem 19.5 (Waldspurger formula). Let f =⊗

fv ∈ π and f =⊗

fv ∈ π. Then

℘H(f)℘H(f)

(f, f)=

1

4·ζF (2)L(1

2, π ⊗ χ)

L(1, η)L(1, π,Ad)︸ ︷︷ ︸L( 1

2,π⊗χ)

·∏v

αv(fv, fv)

L(12, πv ⊗ χv) · (fv, fv)

.

Remark. L(12, π ⊗ χ) may be thought of as the ratio of ℘H(f)℘H(f) by “

∏v αv(fv, fv)” in a

1-dimensional space!54

19.2. Gan–Gross–Prasad conjectures. In the situation of the Waldspurger formula, G =PGL2

∼= SO3 and H = K×/F× ∼= SO2. The Gan–Gross–Prasad conjectures generalize thisto many pairs of classical groups.

The case of interest to us is as follows. Let G = Un = U(V ) and H = Un−1 = U(W ), whereV (resp. W ) is a hermitian space of dimension n (resp. n − 1) with respect to a quadratic

extension K/F . Assume V = W ⊥ K ·u where (u, u) = 1, so that H → G via g 7→(g

1

).

Let π (resp. π′) be a cuspidal tempererd automorphic representation on G (resp. H). Ourgoal is to relate

L

(1

2, πK × π′K

)←→ automorphic H-period ℘H ,

where L(s, πK × π′K) is the Rankin–Selberg L-function for GLn ×GLn−1 from last time.

Conjecture 19.6 (Gan–Gross–Prasad). Write Π = π × π′.(1) (Multiplicity one)

dim HomH(Fv)(Πv,C) ≤ 1.

(2) (ε-dichotomy) ∑(Hi,Π′v)

dim HomHi(Fv)(Π′v,C) = 1,

where (H i,Π′v) runs over the Vogan L-packet of Πv. Moreover, there is a descriptionin terms of ε(Πv).

(3) (Nonvanishing) The following are equivalent:(a) ℘H 6= 0.(b) HomH(A)(Π,C) 6= 0 and L(1

2,Π) 6= 0.

(4) (Ichino–Ikeda formula) Let f ∈ Π and f ′ ∈ Π. Then

℘H(f)℘H(f)

(f, f)=

1

|SΠ|· L(

1

2,Π

)·∏v

αv(fv, fv)

L(12,Πv)(fv, fv)

where SΠ is the component group of the L-parameter of Π.

These are all known:

(1) by Aizenbud–Gourevitch–Rallis–Schiffman for v -∞, and by Sun–Zhu for v | ∞;(2) by Beuzart-Plessis;(3) by W. Zhang, Jacquet–Rallis, Z. Yun, H. Xue, Chaudouard–Zydor;(4) by W. Zhang, Beuzart-Plessis.

20. Lecture 20 (November 28, 2018)

20.1. Bloch–Kato Selmer groups. As motivation, recall that for an elliptic curve E/Qwe have defined the p-Selmer group

Selp(E) ⊆ H1(Q, E[p]),

which sits in a short exact sequence

0→ E(Q)

pE(Q)→ Selp(E)→X(E)[p]→ 0.

55

Selp(E) is cut by local conditions: at v of Q,

im

(δv :

E(Q)

pE(Q)→ H1(Qv, E[p])

).

More generally, we have the p∞-Selmer group

Selp∞(E) := lim−→ Selpn(E) ∼= (Qp/Zp)rp × finite

and the Qp-Selmer group

SelQp(E) := lim←− Selpn(E)⊗Qp = Qrpp .

Our goal is to define an analogue of SelQp(E) for more general Galois representationsρ : GK → Aut(V ) on finite-dimensional Qp-vector spaces V . This will be the Bloch–KatoSelmer group H1

f (K,V ).

Remark. In the case of elliptic curves, V = VpE is the rational Tate module. In particular,SelQp(E) is invariant under isogenies of E.

Let K be a number field.

Definition 20.1. A p-adic Galois representation ρ : GK → Aut(V ) comes from geometry ifV is a subquotient of H i(XK ,Qp)(n) for some i, n, and smooth projective variety X/K.

Remark. Let K be a v-adic local field.

(1) If v - p and X/K has good reduction, then H i(X,Qp) is unramified as a GK-representation.

(2) If v | p, then H i(X,Qp) is de Rham as a GK-representation.

To define de Rham representation, we recall the p-adic comparison theorem.

Theorem 20.2 (Faltings).

H i(X,Qp)⊗Qp BdR∼→ H i

dR(X/K)⊗K BdR,

where BdR is Fontaine’s p-adic period ring.

Using BGKdR = K, we know that

(H i(X,Qp)⊗Qp BdR)GK∼→ H i

dR(X/K).

Definition 20.3. DefineDdR(V ) := (V ⊗Qp BdR)GK .

Say V is de Rham if dimK DdR(V ) = dimQp V .

Returning to the global situation where K is a number field, we make the

Definition 20.4. V is geometric if

(1) V is unramified at almost all places v.(2) V is de Rham at v | p.

So V comes from geometry =⇒ V is geometric. The converse is conjecturally true, sothe condition of V being geometric provides a working definition.

Conjecture 20.5 (Fontaine–Mazur). V comes from geometry if and only if V is geometric.56

Remark. If v | p and X/K has good reduction, then H i(X,Qp) is crystalline at v (definedanalogously using Bcrys instead of BdR).

Theorem 20.6 (Faltings).

H i(X,Qp)⊗Qp Bcrys∼→ H i

crys(Xk/W (k))⊗W (k) BdR,

where k is the residue field of K and W (k) is the Witt ring of k.

Using BGKcrys = W (k)[1

p], the notion of crystalline representations is defined similarly.

Now we are ready to define the local conditions for Bloch–Kato Selmer groups.

Definition 20.7 (Local conditions). Define H1f (Kv, V ) ⊆ H1(Kv, V ) as follows:

(1) If v - p, define the unramified subspace

H1f (Kv, V ) := H1(kv, V

Iv) = ker(H1(Kv, V )→ H1(Iv, V )Frv=1

).

(2) If v | p, define the finite subspace

H1f (Kv, V ) := ker

(H1(Kv, V )→ H1(Kv, V ⊗Qp Bcrys)

).

Similarly to the case of elliptic curves, they satisfy the following properties:

(1) If v - p, thendimH1

f (Kv, V ) = dimH0(Kv, V ).

Also, local duality works: under the local Tate pairing

(·, ·)v : H1(Kv, V )×H1(Kv, V∗(1))→ H2(Kv,Qp(1)) ' Qp,

H1f (Kv, V ) and H1

f (Kv, V∗(1)) are exact annihilators.

(2) If v | p, then

dimQp H1f (Kv, V ) = dimQp H

0(Kv, V ) + dimQp DdR(V )/D+dR(V ),

where the last term computes the number of negative Hodge–Tate weights of V . Also,local duality works.

Example 20.8. Let E/Q be an elliptic curve and V = Vp(E). Then

(1) If v - p, then H1f (Kv, V ) = 0, since the Tate module has no nontrivial Galois invari-

ants. This implies H1(Kv, V ) = 0.(2) If v | p, then dimQp H

1f (Kv, V ) = 1, since HT(V ) = HT(E) = 0,−1. This implies

dimH1(Kv, V ) = 2.

Finally, we can define the Bloch–Kato Selmer groups.

Definition 20.9. The Bloch–Kato Selmer group is defined to be

H1f (K,V ) := s ∈ H1(K,V ) : locv(s) ∈ H1

f (Kv, V ) for all v.

It is a deep theorem that this agrees with the original definition for elliptic curves.

Theorem 20.10 (Bloch–Kato). Let E/K be an elliptic curve (or abelian variety) and V =Vp(E). Then

H1f (Kv, V ) = im

(A(Kv)⊗Qp → H1(Kv, V )

).

Thus H1f (Kv, V ) = SelQp(E).

57

20.2. Bloch–Kato conjecture.

Definition 20.11. Define

L(s, V ) :=∏v-∞

Lv(s, V )

where

Lv(s, V ) =

det(1− Fr−1

v q−sv : V Iv)−1

at v - p,det (1− φ−1

v q−sv : Dcrys,v(V ))−1

at v | p.

Example 20.12. Let E/Q be an elliptic curve. Then

L(s, V ) = L(s+ 1, E),

with center at s = 0.

Conjecture 20.13 (Langlands). Let V be a geometric Galois representation. Then V shouldcome from an automorphic representation. So L(s, V ) has meromorphic continuation tos ∈ C (analytic if V does not contain Qp(1)) and satisfies a functional equation

Λ(s, V ) = ε(s, V )Λ(−s, V ∗(1)).

Conjecture 20.14 (Bloch–Kato).

ords=1 L(s, V ) = dimH1f (K,V ∗(1))− dimH0(K,V ∗(1)).

Here H1f (K,V ∗(1)) is the Bloch–Kato Selmer group, and H0(K,V ∗(1)) = 0 if V is irre-

ducible. The Bloch–Kato conjecture vastly generalizes BSD to any motives.

20.3. Rankin–Selberg motives. Let K/F be a CM extension over a totally real numberfield, and π (resp. π′) be an automorphic representation on GLn,K (resp. GLn−1,K). SetΠ = π × π′.

Our goal is to construct a geometric Galois representation VΠ = Vπ ⊗ Vπ′ , which hasdimension n(n− 1).

Definition 20.15. Let π be a cuspidal automorphic representation on GLn(AK). Say

(1) π is cohomological (or regular algebraic in Henniart’s terminology) if the infinitesimalcharacter χ∞ ∈ X∗(T ) of π∞ is regular, i.e.,

〈χ∞, α∨〉 6= 0

for all α ∈ Φ(GLn). (The Hodge–Tate weights are distinct.)(2) π is conjugate self-dual if πc ∼= π.

Next time we will construct VΠ under these assumptions.

21. Lecture 21 (December 3, 2018)

21.1. BBK for Rankin–Selberg motives. Recall:

Conjecture 21.1 (Bloch–Kato). For a geometric p-adic Galois representation V ,

ords=0 L(s, V ) = dimH1f (K,V ∗(1)).

58

Remark. Suppose K/F is a CM extension of a totally real field, and V c ∼= V ∗(1). Then

L(s, V ) = L(s, V c) = L(s, V ∗(1)),

so we have a functional equation relating

L(s, V )←→ L(−s, V )

with center s = 0.

Now let π be a cuspidal tempered automorphic representation on GLn(AK). The con-struction of Vπ is provided by

Theorem 21.2 (Chenevier–Harris, Shin and many people). Assume π is cohomological andconjugate-self-dual (i.e., πc ' π). Then there exists ρπ : GK → Aut(Vπ) (over a p-adic field)such that:

(1) Vπ is geometric, i.e., de Rham at v | p with distinct Hodge–Tate weights.(2) If πv is unramified, then Vπ is unramified at v if v - p, and cyrstalline at v if v | p. Fur-

thermore, suppose πv corresponds to the Satake parameter Cπv =

µ1

. . .µn

∈GLn(C) with |µi| = 1. Then the eigenvalues of Frv (at v - p) or φv (at v | p) are

given by µiq1−n2 . In other words,

Vπ|Gv = LLC(πv ⊗ | · |

1−n2

v

)under the local Langlands correspondence.

(3) More generally, Vπ|Gv is compatible with LLC(πv).

In particular, we have V cπ = V ∗π (1− n).

Idea of proof. When n is odd, or n is even with π∞ satisfying extra regularity conditions, Vπcan be constructed in the cohomology of a certain unitary Shimura variety, by

(1) point-counting on Igusa varieties (Shin),(2) comparison of Lefschetz trace formula and Arthur–Selberg trace formula.

When n is even, the extra assumption is removed by Chenevier–Harris by congruences(eigenvarieties).

Definition 21.3 (Rankin–Selberg motive). Let π (resp. π′) be an automorphic representa-tion on GLn(AK) (resp. GLn−1(AK)) as before, and Π = π × π′. Define the Rankin–Selbergmotive

VΠ := (Vπ ⊗ Vπ′)(n− 1).

It can be checked easily that V cΠ = V ∗Π(1) and

Lv

(s− 1

2, VΠ

)= det

(1− q−sv Cπv ⊗ Cπ′v

)−1=∏i,j

(1− q−sv µiµ

′j

)−1

for the Satake parameters Cπv and Cπ′v .Now we can state the main theorem, due to Yifeng Liu, Yichao Tian, Liang Xiao, Wei

Zhang and Xinwen Zhu.59

Theorem 21.4 (LTXVV, rank 0). Assume π and π′ have trivial infinitesimal character (withHodge–Tate weights 0,−1,−2, · · · ,−(n− 1) and 0,−1,−2, · · · ,−(n− 2) respectively).Let VΠ be the Rankin–Selberg motive. Then for almost all p,

L

(1

2,Π

)6= 0 =⇒ H1

f (K,V ∗Π(1)) = 0,

i.e., the Bloch–Kato conjecture holds in rank 0.

This is a vast generalization of our previous theorem for elliptic curves.

21.2. Rank one case. We would like to prove the implication

ords= 12L(s,Π) = 1 =⇒ dimH1

f (K,V ∗Π(1)) = 1.

In the elliptic curve case, this crucially relies on the Gross–Zagier formula:

L′(EK , 1) ∼ height of Heegner point yK .

A generalization is given by the arithmetic Gan–Gross–Prasad conjecture:

L′(

1

2,Π

)∼ “height” of GGP cycle ∆Π.

Unfortunately the definition of heights for cycles is conditional on the standard conjectures.Realistically, our goal is to construct ∆Π using the Shimura variety associated to Un×Un−1.

Definition 21.5. Let V be a hermitian space with respect to K/F of dimension n, and Un =U(V ) be the associated unitary group. Assume Un(R) = U(n−1, 1)×U(n, 0)×· · ·×U(n, 0)(with [F : Q] factors in total). Define the Hodge cocharacter

h : C× → Un(R) = U(n− 1, 1)× U(n, 0)× · · · × U(n, 0)

z 7→

zz

1. . .

1

, Id, · · · , Id

and

H := orbit of h under Un(R)-conjugation∼= U(n− 1, 1)/(U(n− 1)× U(1)),

which is a hermitian symmetric domain with dimCH = n− 1.

Write Af = AF,f .

Definition 21.6. Let K ⊆ Un(Af ) be an open compact subgroup. Define the Shimuravariety

ShK(Un)(C) = Un(F )\Un(Af )×H/K.Assume K has no torsion elements. Then ShK is a quasi-projective smooth variety over Fof dimension n− 1.

60

Remark. Technically it is easier to work with the Shimura variety associated to GU(V )instead of U(V ), where

GU(V )(R) = g ∈ GLK⊗R(V ⊗Q R) : 〈gv1, gv2〉 = χ(g)〈v1, v2〉, χ(g) ∈ R×

for each Q-algebra R. There is an extension

1→ U(V )→ GU(V )χ→ Gm → 1.

Consider the Hodge cocharacter

C× → GU(V )(R)

z 7→

z

z. . .

z

, z · Id, · · · , z · Id

.

Then ShK(GU(V )) is of PEL type, with

ShK(GU(V ))(C) = (A, ι, λ, η)

where:

• A is an abelian variety of dimension n− 1,• ι : OK → End(A),

• λ : A∼→ A∨ is an OK-linear polarization,

• η is a K-orbit of isomorphisms

H1(A,Q)⊗Af ' V (Af )

of symplectic Af -modules (up to similitude) and of OK-modules.

Remark. If F 6= Q, then ShK(GU(V )) is projective. If F = Q, then ShK(GU(V )) has acanonical toroidal compactification. Denote them by XV .

Definition 21.7 (GGP cycle). Let V = W ⊕K · u with (u, u) = 1. Then the embeddingGU(W ) → GU(V ) induces δ : XW → XV (of dimensions n − 2 and n − 1 respectively).Consider

(Id, δ) : XW → XW ×XV =: X

(of dimensions n− 2 and 2n− 3 respectively). The GGP cycle is defined as

∆ := im(Id×δ) ⊆ CHn−1(X).

Given Π, define ∆Π = ∆[Πf ] ⊆ CHn−1(X) to be the Πf -isotypic component.

Remark. Local ε-dichotomy ensures there is a unique choice of (W,V ) such that ∆Π ispossibly nonzero.

Conjecture 21.8 (Arithmetic GGP).

L′(

1

2,Π

)∼ “height” of ∆Π.

Remark. The arithmetic GGP is a special case of Beilinson’s conjecture:61

Conjecture 21.9 (Beilinson). Let X/K be a smooth projective variety. Then

ords=center L(s,H2i−1(X)) = dim CHi(X)0,

where CHi(X)0 := ker(CHi(X)cl→ H2i(X)).

Remark. We have the implication

L′(

1

2,Π

)6= 0 =⇒ ∆Π 6= 0.

Then we can consider its image AJ(∆Π) under the Abel–Jacobi map:

CHn−1(X)cl

//

AJ **

H2n−2(X,Qp)(n− 1)[Πf ]

H1(K,H2n−3(XK ,Qp))(n− 1)[Πf ]

where the vertical map comes from the Hochschild–Serre spectral sequence and the fact thatH2n−2(XK ,Qp)(n− 1)[Πf ] = 0. Note that H2n−3 is the exact middle dimension.

Remark. When n = 2, AJ(∆Π) is the analogue of yK under the Kummer map E(K) →H1(K,Vp(E)).

Theorem 21.10 (LTXZZ, rank 1). For almost all p,

AJ(∆Π) 6= 0 =⇒ dimH1f (K,V ∗Π(1)) = 1.

Next time we will sketch the proof of this.

22. Lecture 22 (December 5, 2018)

22.1. Proof strategies. Recall:

Theorem 22.1 (LTXZZ). Let Π be the Rankin–Selberg motive as before.

• (Rank 0) If L(12,Π) 6= 0, then H1

f (K,V ∗Π(1)) = 0.

• (Rank 1) If AJ(∆Π) 6= 0, then dimH1f (K,V ∗Π(1)) = 1.

The strategy is analogous to the elliptic curve case. The key is two explicit reciprocity lawsfor GGP cycles on unitary Shimura varieties. Roughly speaking, for certain “level-raisingprimes” v, we can find a congruence

Π“≡”Π′ (mod p)

where Π is unramified at v and Π′ is ramified at v.The first reciprocity law concerns the case ε(Π) = +1. Recall that the Ichino–Ikeda

formula expresses the central L-value L(12,Π) in terms of the automorphic period ℘Un−1(ϕ)

for ϕ ∈ Π. Then the reciprocity law states that

℘Un−1(ϕ) ≡ ∂v AJ(∆Π′) (mod p),

where AJ(∆Π′) ∈ H1(K,VΠ) with VΠ = H2n−3(X)[Πf ], and ∂v : H1(Kv, VΠ)→ H1sing(Kv, VΠ).

Similarly, we can formulate the second reciprocity law for ε(Π) = −1:

locv AJ(∆Π) ≡ ℘Un−1(Π′) (mod p).

62

Then we use Kolyvagin’s method to bound H1f (K,VΠ).

New ingredients are needed for the explicit reciprocity laws. For X = XV × XW (ofdimensions n− 1 and n− 2 respectively), the Kunneth formula gives

H2n−3(X)[Πf ] = Hn−1(XV )⊗Hn−2(VW )[Πf ],

since cohomology is concentrated at middle degree. Now we have

locv AJ(∆Π) ∈ H1(kv, Hn−1(XV,kv)⊗Hn−2(XW,kv))(n− 1)[Πf ],

so we need to understand the geometry of Shimura varieties in characteristic p.

22.2. Geometry of unitary Shimura varieties at good reduction. Recall that for` - N , X0(N)F` has a distinguished set of supersingular points X0(N)ss

F`. What makes the

explicit reciprocity law work is the identification

X0(N)ssF`' X(N, `)

with the Shimura set associated to B`,∞ (ramified at ` and ∞).

Definition 22.2 (Switching invariants). Let v be a place of F . Assume v is inert in K.Each hermitian space V gives rise to Vv for the local quadratic extension Kv/Fv. Then theHasse invariant is

ε(Vv) :=

+1 if disc(Vv) has even valuation,

−1 if disc(Vv) has odd valuation.

Given V such that ε(Vv) = +1 and sign(V∞) = (n − 1, 1), (n, 0), · · · , (n, 0), define a newhermitian space V • by

ε(V •v ) = −1, sign(V •∞) = (n, 0), (n, 0), · · · , (n, 0).

Theorem 22.3. Assume v is inert in K, and the level defining the unitary Shimura varietyX = XV is given by a hyperspecial subgroup in GU(V )(Fv) (i.e., the stabilizer of a self-duallattice in Vv).

(1) (Kottwitz, Lan) X has an integral model over OKv , given by the solution to a moduliproblem (A, ι, λ, η). The special fiber is smooth projective and has dimension n−1.

(2) (Vollaard–Wedhorn) Let Xss ⊆ Xk be the supersingular locus (i.e., where A/kv is

a supersingular abelian variety). Then the normalization of Xss, denoted by Xss, isa disjoint union of copies of a smooth projective variety DLn of dimension bn−1

2c,

indexed by the definite Shimura set

X• = GU(V •)(Q)\GU(V •)(Af )/K•,

where the level at v for X• is given by the stabilizer ofa self-dual lattice (i.e., Λ∨ = Λ) if n is odd,

an almost self-dual lattice (i.e., Λ∨codim 1

⊇ Λ) if n is even.

63

As a summary, we have the following picture

Xdimn−1

Xss

dimbn−12c

? _oo Xssoo

DLn-fibration

X•dim 0

22.3. Tate’s conjecture. Recall that we have a cycle class map

CHr(X)cl→ H2r(X,Qp)(r).

Conjecture 22.4 (Tate). Assume X/kv is smooth projective over the finite field kv. Then

CHr(X)⊗Qpcl→ H2r(X,Qp)(r)

Frv=1

is surjective.

Remark. This is known when X is an abelian variety and r = 1, due to Tate.

Tate conjecture says that we can understand H2r(X)(r)Fr=1 via algebraic cycles.As before, let π be an automorphic representation on GLn(AK), Vπ be the associated

Galois representation of dimension n, and X = XV be the associated Shimura variety ofdimension n− 1.

Definition 22.5. Assume n is odd. Let v be inert in K. Say v is odd generic for π (mod p)if the eigenvalues α1, · · · , αn of Frv acting on Vπ (mod p) satisfy αi

αj6= 1, qv for all i 6= j.

Theorem 22.6 (Xiao–Zhu). Assume v is odd generic. Then Tate conjecture holds for

Hn−1(Xkv)(n−1

2)[πf ]

Frv=1. In fact, this space is exactly generated by Xss.

In particular, this and Poincare duality together imply that

Hn−1(Xkv)(n−1

2)[πf ]

Frv=1 ∼→ Hn−1(Xss)(n−12

)[πf ]

' H0(X•)[πf ]∗,

which is a space of automorphic forms!

22.4. Second reciprocity.

Definition 22.7. Let n be even. Say v is even level-raising for π (mod p) if αiαj6= 1, qv for

all i 6= j, except that αn2

= qn2v and αn

2+1 = q

n2−1

v .

This is analogous to the Steinberg representation for GL2.

Definition 22.8. Say p is nice if p is a good prime for Π = π × π′, p ≥ 5, and:

(1) Vπ and Vπ′ mod p are irreducible and with adequate images;(2) p is isolated for Π.

Fact. p is nice for almost all p.

Definition 22.9. Assume n is even. Say v is admissible for Π = π× π′ (on GLn and GLn−1

respectively) mod nice p if:64

(1) v is odd generic for π′;(2) v is even level-raising for π;(3) Frv acting on VΠ has eigenvalue 1 with multiplicity 1.

Fact. v is admissible for a positive proportion of v.

Theorem 22.10 (Second explicit reciprocity). Assume p is nice and v is admissible. Iflocv AJ(∆Π) 6≡ 0 (mod p), then there exists Π• on GU(V •) × GU(W •) and ϕ• ∈ Π• suchthat

locv AJ(∆Π) ≡ ℘GU(W •)(ϕ•) (mod p).

23. Lecture 23 (December 10, 2018)

23.1. Tate conjecture via geometric Satake. Recall that a new ingredient in provingexplicit reciprocity is the Tate conjecture for unitary Shimura varieties. Let X = XV,k bethe special fiber associated to GU(1, n−1). Assume n is odd, so that dimX = n−1 is even.

Last time we saw the diagram

Xdimn−1

Xss

dim n−12

oo

X•dim 0

Shimura set

where the fibers of the vertical map are Deligne–Lusztig varieties of dimension n−12

.

Theorem 23.1 (Xiao–Zhu). Assume v is odd generic for π. Then

Hn−1(Xss)(n−12

)[πf ] ' Hn−1(X)(n−12

)Frv=1[πf ].

Remark. This even holds with coefficients Λ = Fp. Let mπ ⊆ T correspond to π. Then

Hn−1(Xss,Λ)(n−12

)mπ∼→ Hn−1(X,Λ)(n−1

2)Frv=1mπ .

This follows from the torsion-free result of Caraiani–Scholze.

The proof strategy is:

(1) (Injectivity) Show the image of components of Xss are linearly independent inHn−1(X).(2) (Compare dimensions) Prove dimensions are equal using comparison of Lefschetz

trace formula for X• and X.

(1) is more difficult: by looking at the intersection pairing

(·, ·) : Hn−1(X)×Hn−1(X)→ H2(n−1)(X) ' Qp,

it suffices to show the matrix under (·, ·) for Hn−1(Xss) is nondegenerate. This intersectionmatrix is very difficult to compute, and the idea is to compute the determinant abstractly

using the representation theory of G.

• On the generic fiber, we have an action of Hecke correspondence Hv at v. For Gsplit, the Satake isomorphism gives

Hv∼→ C[X∗(T )]W

∼→ C[G]G.65

• On the special fiber, we can realize

Xss

X• X

as an “exotic Hecke correspondence” between X• and X. This has meaning in terms

of “spectral operators”, which comes from the representation theory of G.

23.2. Geometric Satake. Let F be a local non-archimedean field with residue field k andring of integers O = OF ⊆ F . Let G/O be a reductive group.

Definition 23.2. The spherical Hecke algebra is H := C∞c (G(O)\G(F )/G(O)).

The Satake isomorphism states that

H ∼→ Qp[G]G

with an action by σ-conjugation: for σ the Frobenius, g · h = ghσ(g−1).

Remark. We can rewrite this as H ∼→ Qp[Gσ]G, where Gσ ⊆ Go 〈σ〉.

To geometrize the classical Satake isomorphism, we need to introduce:

Definition 23.3. The local Hecke stack is

Hkloc := L+G\LG/L+G,

where LG(k) = G(F ) and L+G(k) = G(O). For every perfect k-algebra R, define the diskDR = SpecWO(R) and the punctured disk D∗R = SpecWO(R)[ 1

$]. Then

Hkloc(R) = E1β→ E2 : E1 and E2 are G-bundles on DR and β|D∗R is an isomorphism.

Let Perv(Hkloc) be the category of perverse sheaves on Hkloc.

Theorem 23.4 (Mirkovic–Vilonen if charF > 0; X. Zhu if charF = 0). There is anequivalence of monoidal categories

Rep(G)∼→ Perv(Hkloc)

Vµ 7→ ICµ

which sends the highest weight representation of weight µ to the IC sheaf on Grµ ⊆ Gr.

23.3. Spectral operators.

Definition 23.5. The moduli of local shtukas is

Shtloc := (E1β→ E2) ∈ Hkloc with σ∗E1

∼→ E2.

By definition, we have a forgetful map Shtloc → Hkloc.66

Definition 23.6. Define

Hk(Shtloc) :=

E1

β//

α

E2

σ∗α

∼// σ∗E1

E ′1β′// E ′2

∼// σ∗E ′1

.

Then there is a diagram

Hk(Shtloc)←−h

yy

−→h

%%

Shtloc Shtloc

Let Shtlocµ ⊆ Shtloc be given by the locus where inv(β) ≤ µ, and Shtloc

µ1|µ2 ⊆ Hk(Shtloc) bewhere inv(β) ≤ µ1 and inv(β′) ≤ µ2. Then we have a correspondence

Shtlocµ1|µ2

##

Shtlocµ1

Shtlocµ2

Example 23.7.

• Shtloc0 = [Spec k/G(O)] = [point/G](k) (where G(O) is considered as a discrete

group).• Shtloc

0|0 = [G(O)\G(F )/G(O)] = Hkloc(k).

Definition 23.8. The category PervCorr(Shtloc) is defined by:

(1) objects: perverse sheaves on Shtloc;(2) morphisms: cohomological correspondences supported on Hk(Shtloc), i.e.,

HomCorr(F1,F2) := Hom(←−h ∗F1,

−→h !F2).

The identification Rep(G) = Coh([•/G]) induces a pullback map

Rep(G)→ Coh([Gσ/G])

V 7→ V := V ⊗OGσ

with image inside the subcategory Cohfree([Gσ/G]) of free sheaves.

Theorem 23.9 (Xiao–Zhu). There is a functor S such that the diagram

V_

Rep(G)∼

Satake//

pullback

Perv(Hkloc)

pullback

V = V ⊗OGσ Cohfree([Gσ/G])∃S// PervCorr(Shtloc)

commutes.67

Remark. Let V0 = 1 be the trivial representation of G. Then

End(V0) = Γ([Gσ/G],O) = Qp[Gσ]G(

Satake' H)

is the space of twisted conjugation-invariant functions on G. More generally, define

J(V ) = Γ([Gσ/G], V )

= (Qp[Gσ ⊗ V ])G

= f : G→ V : f(ghσ−1(g)) = g · f(h).

Write J := J(V0) (' H).

23.4. Computing intersection numbers. Notice there is a map X = Sh(G, µ) → Shtlocµ

and a diagram

Xss

##

X•

Shtloc0|µ

##

X

Shtloc0 Shtloc

µ .

Then Xss can be identified via

H0(X•)⊗J

Hom(V0, Vµ)→ Hn−1(X).

Here Hom(V0, Vµ) = J(Vµ) can be thought of as an “exotic correspondence” via the S-operator. Finally, the intersection matrix is given by

J(Vµ)× J(V ∗µ )→ J.

Returning to the case G = U(n) for n odd, µ(t) =

t

1. . .

1

and Vµ is the standard

representation of GLn, the upshot is:

Theorem 23.10.

(1) J(Vµ) is a free module of rank 1 over J .(2) The intersection matrix (1× 1) is given by

n−12∏i=1

(Xi +X−1i − 2),

where Xi(t) = diag(1, · · · , t, · · · , 1, · · · , 1, · · · , t−1, · · · , 1) with t (resp. t−1) at thei-th (resp. (n− 1− i)-th) position.

68

Corollary 23.11. After localizing at mπ, we getn−12∏i=1

(µi + µ−1i − 2)

where µi are the Satake parameters of π. This is nonzero because µi 6= 1!

69