darmon points for fields of mixed signature

Darmon points for fields of mixed signatureFirst Joint International Meeting 2014

Bilbo, July 2nd, 2014

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Institut fur Experimentelle Mathematik

2,3University of Warwick

Marc Masdeu Darmon points for fields of mixed signature 0 / 19

The Hasse-Weil L-function

Let F be a number field (assume h+F = 1).

Let E/F be an elliptic curve of conductor N = NE .Let K/F be a quadratic extension of F .

I Assume that N is square-free, coprime to disc(K/F ).

L-function of E/K (for <(s) >> 0)

L(E/K, s) =∏p|N

(1− ap|p|−s

)−1 ×∏p-N

(1

ap(E) = 1 + |p| −#E(Fp).

− ap|p|−s + |p|1−2s)−1.

Assume that E is modular =⇒I Analytic continuation of L(E/K, s) to C.I Functional equation relating s↔ 2− s.


The BSD conjecture

Bryan Birch Sir Peter Swinnerton-Dyer

BSD conjecture (coarse version)

ords=1 L(E/K, s) = rkZE(K).

So L(E/K, 1) = 0BSD=⇒ ∃PK ∈ E(K) of infinite order.

I How to construct such PK?


Goals of this talk

1 Sketch a general construction of Darmon points.2 Explain some algorithmic challenges we face in their computation.

“ The fun of the subject seems to me to be in the examples.B. Gross, in a letter to B. Birch, 1982”

3 Illustrate with an example.


Our construction: Highlights

Available for arbitrary base number fields F (mixed signature).I We assume h+F = 1 “for simplicity”.

Comes in both archimedean and non-archimedean flavors.I The Heegner point construction also has a (less known)

non-archimedean counterpart!All of the previously constructions become particular cases:

1 Archimedean:F Darmon (Rational points on modular elliptic curves, 2002)F Gartner (Canad. J. Math., 2012)

2 Non-archimedean:F Darmon (Ann. of Math. (2), 2001)F Trifkovic (Duke Math. J., 2006)F Greenberg (Duke Math. J., 2009)


Goals of this talk





Basic notationRecall: if v | ∞F is an infinite place of F , then:

I If v is real, then:1 It may extend to two real places of K (splits), or2 It may extend to one complex place of K (ramifies).

I If v is complex, then it extends to two complex places of K (splits).

n = #v | ∞F : v splits in K.

K/F is CM ⇐⇒ n = 0.

S(E,K) =v | N∞F : v not split in K

, s = #S(E,K).

S(E,K) has already appeared this week:I S(E,K) = ∞Q in Vigni’s talk (Heegner hypothesis)I S(E,K) = v0 in Guitart’s talk.

Sign of functional equation for L(E/K, s) should be (−1)#S(E,K).I From now on, we assume that s is odd.

In particular, S(E,K) is nonempty!I In this talk: assume S(E,K) contains some finite prime.


Non-archimedean path integralsFor simplicity: Suppose that |p| = p (totally split prime).Hp = P1(Cp) r P1(Qp) has a rigid-analytic structure.SL2(Qp) acts on Hp through fractional linear transformations:(

a bc d

)· z =

az + b

cz + d, z ∈ Hp.

Use rigid-analytic 1-forms ω ∈ Ω1Hp .

Coleman integral to integratebetween τ1 and τ2 in Hp.Compatibility with the action of SL2(Qp)on Hp: ∫ γτ2

γτ1

ω =

∫ Q

Pγ∗ω.

Robert ColemanMarc Masdeu Darmon points for fields of mixed signature 7 / 19

Coleman Integration on Hp

ω ∈ Ω1Hp completely determined by its “restriction to the boundary”:

I ∂Hp can be identified with P1(Qp).I The 1-form ω gives rise to a “boundary measure” µω on P1(Qp).

Coleman integration on Hp can be defined as:∫ τ2

τ1

ω =

∫P1(Qp)

logp

(t− τ2

t− τ1

)dµω(t) = lim

U

∑U∈U

logp

(tU − τ2

tU − τ1

)µω(U).

I If µω(U) ∈ Z for all U , have a multiplicative refinement.

Bruhat-Tits tree of GL2(Qp).Hp having the Bruhat-Tits as retract.Opens for a covering of size p−3.tU is any point in U ⊂ P1(Qp).

P1(Qp)

U ⊂ P1(Qp)


The group Γ

Choose a (finite) prime p ∈ S(E,K).Let B/F = quaternion algebra with Ram(B) = S(E,K) r p.

I B = M2(F ) (split case) ⇐⇒ s = 1.

Let D be the discriminant of B (product of finite ramified primes).Let m be such that NE = pDm.Let RD

0 (m) be an Eichler order of level m inside B.Fix an embedding ιp : RD

0 (m) →M2(ZF,p).

Γ = RD0 (m)[1/p]×1

ιp→ SL2(Fp).

e.g. S(E,K) = p gives

Γ =(

a bc d

)∈ SL2(OF [1/p]) : c ∈ m[1/p]

.

Set ΓD0 (m) = RD

0 (m)×1 .


Overview of the constructionWe attach to E a cohomology class

ΦE ∈ Hn(Γ,Ω1Hp,Z).

We attach to each embedding ψ : K → B a homology class

Θψ ∈ Hn

(Γ,Div0Hp

).

I Well-defined up to the image of Hn+1(Γ,Z)δ→ Hn(Γ,Div0Hp).

Cap-product and integration on the coefficients yield an element:

Jψ = ×∫

Θψ

ΦE ∈ K×p .

Jψ is well-defined up to a multiplicative lattice

L =×∫δ(θ)

ΦE : θ ∈ Hn+1(Γ,Z).


Conjecture 1 (Greenberg, Guitart-M-Sengun)

There is an isogeny β : K×p /L→ E(Kp).

Dasgupta–Greenberg, Rotger–Longo–Vigni: some non-arch. cases.

Jψ = ×∫

Θψ

ΦE ∈ K×p /L.

The Darmon point attached to E and ψ : K → B is:

Pψ = β(Jψ) ∈ E(Kp).

Conjecture 2 (Darmon, Greenberg, Trifkovic, G-M-S)1 The local point Pψ is global, and belongs to E(Kab).2 Pψ is nontorsion if and only if L′(E/K, 1) 6= 0.

Predicts also the exact number field over which Pψ is defined.Understand the Galois action on Pψ in terms class field theory.


HomologyLet ψ : O → RD

0 (m) be an embedding of an order O of K.I Which is optimal: ψ(O) = RD

0 (m) ∩ ψ(K).Consider the group O×1 = u ∈ O× : NmK/F (u) = 1.

I rank(O×1 ) = rank(O×)− rank(O×

F ) = n.

Choose a basis u1, . . . , un ∈ O×1 for the non-torsion units.I ; ∆ψ = ψ(u1) · · ·ψ(un) ∈ Hn(Γ,Z).

K× acts on Hp through K×ψ→ B×

ιp→ GL2(Qp).

I Let τψ be the (unique) fixed point of K× on Hp.Have the exact sequence

Hn+1(Γ,Z)δ // Hn(Γ,Div0Hp) // Hn(Γ,DivHp)

deg // Hn(Γ,Z)

Θψ ? // [∆ψ ⊗ τψ] // [∆ψ]

Fact: [∆ψ] is torsion.I Can pull back a multiple of [∆ψ ⊗ τψ] to Θψ ∈ Hn(Γ,Div0Hp).I Well defined up to δ(Hn+1(Γ,Z)).


Goals of this talk





Computing in practice (n = 1)Homology: The main problem is to lift the cycle:

H2(Γ,Z)δ // H1(Γ,Div0Hp) // H1(Γ,DivHp)

deg // H1(Γ,Z)

Θψ ? // [∆ψ ⊗ τψ] // [∆ψ]

I Use finite presentation + word problem for Γ (Voight, Page).Cohomology: Use a generalization of modular symbols.

I More interesting for higher n (more geometry).

Integration: Could use the definition:∫ τ2

τ1

ω =

∫P1(Qp)

logp

(t− τ2

t− τ1

)dµω(t) = lim

U

∑U∈U

logp

(tU − τ2

tU − τ1

)µω(U).

(1)

I Yields exponential algorithm: one p-adic digit requires p times morework.


Overconvergent Method (I)

We have attached to E a cohomology class Φ ∈ H1(Γ,Ω1Hp,Z).

Goal: to compute integrals∫ τ2τ1

Φγ , for γ ∈ Γ.Recall that ∫ τ2

τ1

Φγ =

∫P1(Qp)

logp

(t− τ1

t− τ2

)dµγ(t).

Write P1(Qp) = |z| > 1 ∪⋃p−1a=0(a+ pZp).

Expand the integrands into power series and change variables.I We reduce to calculating the moments:∫

Zp

tidµγ(t) for all γ ∈ Γ, for all i ≥ 0.

Recall ΓD0 (m) = RD

0 (m)×1 , and note that Γ ⊇ ΓD0 (m) ⊇ ΓD

0 (pm).


Overconvergent Method (II)D = locally analytic Zp-valued distributions on Zp.

I ϕ ∈ D maps a locally-analytic function h on Zp to ϕ(h) ∈ Zp.I D is naturally a ΓD

0 (pm)-module.

The map ϕ 7→ ϕ(1Zp) induces a projection:

ρ : H1(ΓD0 (pm),D)→ H1(ΓD

0 (pm),Zp).

Recall that ΦE comes (Shapiro’s lemma) from ϕE ∈ H1(ΓD0 (pm),Zp):

ϕE(γ) =

∫Zpµγ(t).

Theorem (Pollack-Stevens, Pollack-Pollack)

There exists a unique Up-eigenclass Φ lifting ϕE .

Moreover, Φ is explicitly computable by iterating the Up-operator.


Overconvergent Method (III)

But we needed to compute the moments of a system of measures!

PropositionConsider the map Ψ: ΓD

0 (pm)→ D:

γ 7→ Ψ(γ), Ψ(γ)(h(t)) =

∫Zph(t)dµγ(t).

1 Ψ induces a class [Ψ] ∈ H1(ΓD0 (pm),D).

2 [Ψ] is a lift of µ.3 [Ψ] is a Up-eigenclass.

Corollary

The algorithmically computed Φ “knows about” the above integrals.


Where isthe example ??


Non-archimedean cubic Darmon point (I)

F = Q(r), with r3 − r2 − r + 2 = 0.F has signature (1, 1) and discriminant −59.Consider the elliptic curve E/F given by the equation:

E/F : y2 + (−r − 1)xy + (−r − 1) y = x3 − rx2 + (−r − 1)x.

E has conductor NE =(r2 + 2

)= p17q2, where

p17 =(−r2 + 2r + 1

), q2 = (r) .

Consider K = F (α), where α =√−3r2 + 9r − 6.

The quaternion algebra B/F has discriminant D = q2:

B = F 〈i, j, k〉, i2 = −1, j2 = r, ij = −ji = k.


Non-archimedean cubic Darmon point (II)The maximal order of K is:

OK = OF +wKOF , wK satisfying x2 + (r+ 1)x+7r2 − r + 10

16= 0.

One can embed OK in the Eichler order of level p17 by:

wK 7→ ψ(wK) = (−r2 + r)i+ (−r + 2)j + rk.

We obtain γψ = ψ(u) = 6r2−72 + 2r+3

2 i+ 2r2+3r2 j + 5r2−7

2 k, and

τψ = (12g+8)+(7g+13)17+(12g+10)172+(2g+9)173+(4g+2)174+· · ·

After integrating we obtain:

Jψ = 16+9·17+15·172+16·173+12·174+2·175+· · ·+5·1720+O(1721),

which corresponds to:

Pψ = −108 ·(r − 1,

α+ r2 + r

2

)∈ E(K).


Eskerrik Asko !

Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/


http://www.warwick.ac.uk/mmasdeu/

BibliographyH. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves.Int. Math. Res. Not. (2003), no. 40, 2153–2180.

H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: Algorithms and evidence.Exp. Math., 11, No. 1, 37-55, 2002.

H. Darmon and R. Pollack. Efficient calculation of Stark-Heegner points via overconvergent modular symbols.Israel J. Math., 153:319–354, 2006.

J. Gartner. Darmon points and quaternionic Shimura varieties.Canad. J. Math. 64 (2012), no. 6.

X. Guitart and M. Masdeu. Elementary matrix Decomposition and the computation of Darmon points with higher conductor.Math. Comp. (arXiv.org, 1209.4614), 2013.

X. Guitart and M. Masdeu. Computation of ATR Darmon points on non-geometrically modular elliptic curves.Exp. Math., 2012.

X. Guitart and M. Masdeu. Computation of quaternionic p-adic Darmon points.(arXiv.org, 1307.2556), 2013.

X. Guitart, M. Masdeu and M.H. Sengun. Darmon points on elliptic curves over number fields of arbitrary signature.(arXiv.org, 1404.6650), 2014.

M. Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties.Duke Math. J., 147(3):541–575, 2009.

D. Pollack and R. Pollack. A construction of rigid analytic cohomology classes for congruence subgroups of SL3(Z).Canad. J. Math., 61(3):674–690, 2009.

M. Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields.Duke Math. J., 135, No. 3, 415-453, 2006.


Aside: an interesting by-productLet Φ ∈ Hn

(Γ,Ω1

Hp

)be an eigenclass with integer eigenvalues.

In favorable situations Φ “comes from” an elliptic curve E over F .No systematic construction of such curves for non totally real F .We can compute the lattice

L =×∫δ(θ)

Φ: θ ∈ Hn+1(Γ,Z)unram. quadratic ext. of Fν .

⊂ F×ν2.

Suppose that Conjecture 1 is true.From L one can find a Weierstrass equation Eν(Fν2) ∼= F×

ν2/L.

I Hopefully the equation can be descended to F .

A similar technique (in the archimedean case) used by L. Dembeleto compute equations for elliptic curves with everywhere goodreduction.Stay tuned!


Available Code

SAGE code for non-archimedean Darmon points when n = 1.

https://github.com/mmasdeu/darmonpoints

I Compute with “quaternionic modular symbols”.F Need presentation for units of orders in B (J. Voight, A. Page).

I Implemented overconvergent method for arbitrary B.I We obtain a method to find algebraic points.

SAGE code for archimedean Darmon points (in restricted cases).

https://github.com/mmasdeu/atrpoints

I Only for the split (B = M2(F )) cases, and:1 F real quadratic, and K/F ATR (Hilbert modular forms)2 F cubic (1, 1), and K/F totally complex (cubic automorphic forms).


https://github.com/mmasdeu/darmonpoints

https://github.com/mmasdeu/atrpoints

Cycle Decomposition: exampleG = R×1 , R maximal order on B = B6.

F = 〈X,Y 〉 G = 〈x, y | x2 = y3 = 1〉.

Goal: write g ⊗ τ as∑gi ⊗Di, with Di of degree 0.

Take for instance g = yxyxy. Note that wt(x) = 2 and wt(y) = 3.First, trivialize in Fab: g = yxyxyx−2y−3.To simplify γ ⊗ τ0 in H1(Γ,DivHp), use:

1 gh⊗D ≡ g ⊗D + h⊗g−1D.2 g−1 ⊗D ≡ −g ⊗gD.

g ⊗ τ0

= yxyxyx−2y−3 ⊗ τ0

= y ⊗ τ0

+ xyxyx−2y−3 ⊗y−1τ0 = y ⊗ τ0

+ xyxyx−2y−3 ⊗ τ1 = y ⊗ τ0 + x⊗ τ1

+ yxyx−2y−3 ⊗x−1τ1 = y ⊗ τ0 + x⊗ τ1

+ yxyx−2y−3 ⊗ τ2 = y ⊗ τ0 + x⊗ τ1

+ y ⊗ τ2 + xyx−2y−3 ⊗y−1τ2 = y ⊗ (τ0 + τ2) + x⊗ τ1

+ xyx−2y−3 ⊗ τ3 = y ⊗ (τ0 + τ2) + x⊗ τ1

+ x⊗ τ3 + yx−2y−3 ⊗x−1τ3 = y ⊗ (τ0 + τ2) + x⊗ (τ1 + τ3)

+ yx−2y−3 ⊗ τ4 = y ⊗ (τ0 + τ2) + x⊗ (τ1 + τ3)

+ y ⊗ τ4 + x−2y−3 ⊗y−1τ4 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

+ x−2y−3 ⊗ τ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

+ x−1 ⊗ τ5 + x−1y−3 ⊗xτ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

− x⊗xτ5 + x−1y−3 ⊗xτ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

− x⊗ τ6 + x−1y−3 ⊗ τ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

+ x−1 ⊗ τ6 + y−3 ⊗xτ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

− x⊗xτ6 + y−3 ⊗xτ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

− x⊗ τ7 + y−3 ⊗ τ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−3 ⊗ τ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ7 + y−2 ⊗yτ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ7 + y−2 ⊗yτ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ8 + y−2 ⊗ τ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−2 ⊗ τ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ8 + y−1 ⊗yτ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ8 + y−1 ⊗yτ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ9 + y−1 ⊗ τ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ10 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9 − τ10) + x⊗ (τ1 + τ3 − τ6 − τ7)


darmon points for fields of mixed signature

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