analytic construction of points on modular elliptic curves

Analytic construction of pointson modular elliptic curves

Number Theory Study GroupUniversity of York

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Universitat de Barcelona

2University of Warwick

3University of Sheffield

May 26, 2015

Marc Masdeu Analytic construction of pointson modular elliptic curves May 26, 2015 0 / 23

Elliptic Curves

Let E be an elliptic curve defined over Q:

Y 2 + a1Y + a3XY = X3 + a2X2 + a4X + a6, ai ∈ Z

Let K/Q be a number field, and consider the abelian group

E(K) = (x, y) ∈ K2 : y2 +a1y+a3xy = x3 +a2x2 +a4x+a6∪O.

Theorem (Mordell–Weil)E(K) is finitely generated: E(K) ∼= (Torsion)⊕ Zr.

The integer r = rkZE(K) is called the algebraic rank of E(K).I Open problem: Given E and K, find r.

Theorem (Weierstrass Uniformization)There exists a computable complex-analytic group isomorphism

η : C/ΛE → E(C), ΛE = lattice of rank 2.Marc Masdeu Analytic construction of pointson modular elliptic curves May 26, 2015 1 / 23

The Hasse-Weil L-function

Suppose that K = Q(√D) is quadratic.

Can reduce the coefficients a1, . . . , a6 modulo primes p.I For almost all primes, the reduction is an elliptic curve (nonsingular).I Obviously E(Fp) is finite.

The conductor of E is an integer N encoding the shape of E whenthis reduction is singular.

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

The L-function of E/K (Re(s) > 3/2)

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.

Modularity (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor)=⇒

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.

Open problem: construct such PK .Marc Masdeu Analytic construction of pointson modular elliptic curves May 26, 2015 3 / 23

Modular forms

Let N > 0 be an integer and consider

Γ0(N) = (a bc d

)∈ SL2(Z) : N | c.

Recall the upper-half plane H = z ∈ C : Im(z) > 0.I Γ0(N) acts on H via

(a bc d

)· z = az+b

cz+d .A cusp form of level N is a holomorphic map f : H→ C such that:

1 f(γz) = (cz + d)2f(z) for all γ =(a bc d

)∈ Γ0(N).

2 Cuspidal: limz→i∞ f(z) = 0.Since ( 1 1

0 1 ) ∈ Γ0(N), have Fourier expansions

f(z) =∞∑n=1

an(f)e2πinz.

The (finite) vector space of all cusp forms is denoted by S2(Γ0(N)).There is a family of commuting linear operators (Hecke algebra)acting on S2(Γ0(N)), indexed by integers coprime to N .

I A newform is a simultaneous eigenvector for the Hecke algebra.


Modularity

Theorem (Modularity, automorphic version)Given an elliptic curve E, there exists a newform fE ∈ S2(Γ0(N)) s.t.

ap(fE) = 1 + p−#E(Fp), for all p - N.

The complex manifold Y0(N)(C) = Γ0(N)\H can be compactified byadding a finite set of points (cusps), yielding X0(N)(C).Shimura proved that X0(N)(C) is the set of C-points of an algebraic(projective) curve X0(N) defined over Q.

Theorem (Modularity, geometric version)Given an elliptic curve E, there exists a surjective morphism

φE : X0(N)/Q→ E.


The main tool for BSD: Heegner points

Kurt Heegner

Only available when K = Q(√D) is imaginary: D < 0.

I will define Heegner points under the additional condition:I Heegner hypothesis: p | N =⇒ p split in K.

This ensures that ords=1 L(E/K, s) is odd (so ≥ 1).Modularity is crucial in the construction.


Heegner Points (K/Q imaginary quadratic)

Modularity =⇒ modular form fE attached to E.

ωE = 2πifE(z)dz = 2πi∑n≥1

an(f)e2πinzdz ∈ Ω1Γ0(N)\H.

I Can be obtained simply by counting points of E(Fp) (very efficient).

Given τ ∈ K ∩H, set Jτ =

∫ τ

i∞ωE ∈ C.

Well-defined up to ΛE =∫

γ ωE | γ ∈ H1 (X0(N),Z)

.

Set Pτ = η(Jτ ) ∈ E(C).Fact: Pτ ∈ E(Hτ ), where Hτ/K is a ring class field attached to τ .

Theorem (Gross–Zagier, Kolyvagin)1 PK = TrHτ/K(Pτ ) is nontorsion ⇐⇒ L′(E/K, 1) 6= 0.2 If ords=1 L(E/Q, s) ≤ 1 then BSD holds for E(Q).


Computing in practice: an example of Mark WatkinsLet E be the elliptic curve of conductor NE = 66157667:

E : Y 2 + Y = X3 − 5115523309X − 140826120488927.

Watkins worked with 460 digits of precision and 600M terms of theL-series. Took less than a day (in 2006). The x-coordinate of the pointhas numerator:3677705371866775066140056423418271700879322694922855847262187700616535463492710158053651343703267430611413064645000528867046519983997664788407919153078617415072739338026281573250924797082687602171017553858718167805487654785022844156276828471927526818990949626599378706300367603592935770218062374839710749312284163465078523816968832276500720399644815972159959932997449341171062898503893640065524978358777402575345331137752028822100483561636459193457948120745710296608971732243703377010561657350085906402970902987091215062666972664619932018253973699995508681422943127563221774107305328280647596049753692423509935680307269370499116072641097827468479512837941192989412144907943309029865829912295694015235199387427463761071907702040105138183490127866378892547110594555551738109049119276198990318551492923253385898319797370264027110497425941160003806014808399829755575060358517280356452410442291650296493470492891191885968694011593251313633459625795031323398472754224400945538247051892256536774595128631179117218385529343091245081344933664374080939243620397499119074169735041423221117570585842007250226321161647201649986417295226774605259994990779421258204288795260637356926859910185168629387960475973239865371541712483169437963732171919939969937146546295368843960579247909386476566632815961781457221160982165009303338243218067269370181901361905565732088070483553355670787931266569286578590367793505932745987173797308807240343018677394437498418094567158841937203289014615526598826284058422097567571678166621399450818646421085335959899757162592592401528340509406544796171476859225008569444498220453860921224090969785448172188478976405134778065983291776042463808123777390491844755507773416209859765703930378802827649670195524084007307548226764414817153853440019798322326524148883358655673772143604560032969616681774819448090662574425967723478296641269729319041016852811289447800746467967609424309596170222574798740894035649650388853798178669200489298145202684936775070590737659026716380873664884967028363262685745931232451074203488781017631238933476570202755912488242478005942708620520821859733932900091898676772594580806760650987034535395255769756395437005076256407298723407894063143944684005844559206833619762001218344307512339014732284974905619980784862510749935288713187974033480873704269009975564425770812549105721851078566051398773310150428421211060806907435781732684894004990568983126219539479670123584145477528970810970917957442036976840460662556632012422927601267598712660045163774329619172720402171470835633999876124205952757920338556769918233682548621595584500438080514815332972700352873822470382792932239463850701180823069589872686033969240544031038574440588486055874154005176700326311212061277324813403910882777964885444157381565530147684062461546660051396904280851450982725007914162147746734845018267225005270911649442625371695958489316807540967747128604905727462240940311870432045261072392010796034682975228951065985674370150833487978753641627976939688198041395488857512826871522370782603587052302844262030644936842506142828799181077337962070672500038239594129356776240932360470386373655773263995890088045077860119731559277310730347065365574614438066227076224110878093718721572104568368924936138367920267618203822171654819989241236047827879232297391719205754470070995016783807950770131133259898013857299939208183016544242513395646068768201219283722462133998592132827925111680439534438397939011399741944793002975660976645391993846519084361887324288183733023830463888594279378938418880142666851776166056447837041357949318307502656863359340665652409440494482130055919971289855607602603992142786359126343515867623548693540215307461899928995825545976321083096385692969648000469830727362384831490147146008960565520296427479914190634547491420595642742982546549258938664049551469033300244757461635437149962496524201711710542317263364935415869714317789440514810596337383994114185743238117709497297268436126729250006313556598341642005544413154510034334524662047071238116636236628372968629480617587599286317636619851856158018862057707210320063041448677873470583163922956715800916558720872094859132869301288586404425891254542685803974845719210123188723116248983176156076281764600974413363235490318282359656362779508273280875479395111123742164365842033792484501226474060940351711307406637235476759398859593638811358930351020183894442127461462503283482426106735240223789949783920200988147219745020626928157366892297590658220939427953187053452755989894263352359355056053114113015603211922694308617337435444029085864973053536009094312149332025225287171092144929593300160658102876231441792884666648885406227023467042137524563725744495639792157824065669378853529458719945417708388719305422203077716714984665181087226221094216767415449456954035098669531672776282802324648392150034740488969680375446600297557400655812701390832499032125722304179422497954671007003939443103250096771791821099709433468073350144468396122825088243240736795841228512083604591663154848919522994493400258965092989359393577217235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Heegner Points: revealing the trick

Why did this work?

1 The Riemann surface Γ0(N)\H has an algebraic model X0(N)/Q.

2 Existence of the morphism φE defined over Q:

φE : X0(N)→ E. (geometric modularity)

3 CM theory shows that τ ∈ Γ0(N)\H are defined in X0(N)(Hτ ).An explicit description of φ shows that:

Pτ = φ(τ) ∈ E(Hτ ).


Generalization

One can replace Q with any totally real field F , m = [F : Q].I i.e. the defining polynomial of F factors completely over R.

Consider an elliptic curve E defined over F , of conductor NE .The field K/F needs then to be a CM extension.

I i.e. the defining polynomial for K over Q has no linear terms over R.

Suppose that NE is coprime to the discriminant of K/F .Let Sf (E,K) = p | NE : p inert in K.The Heegner hypothesis can be relaxed to:

(new) Heegner Hypothesis: m+ #Sf (E,K) is odd.

I This still ensures that ords=1 L(E/K, s) is odd.I Old Heegner hypothesis is equivalent to Sf (E,K) = ∅ (and m = 1).


Darmon’s Idea

Henri Darmon

What if K/F is not CM?

I Simplest case: F = Q, K real quadratic.

I Or what if F is not totally real?


However. . .

. . . this will get us in trouble!

1 Algebraic model X/F .

2 Geometric modularity: φE : X → E.

3 CM points τ ∈ X(Hτ ).


History

Henri Darmon Adam Logan Xevi Guitart Jerome Gartner

H. Darmon (2000): F totally real, Sf (E,K) = ∅.I Darmon-Logan (2003): F quadratic norm-euclidean, NE trivial.I Guitart-M. (2011): F quadratic norm-euclidean, NE trivial.I Guitart-M. (2012): F quadratic norm-euclidean, NE trivial.

J. Gartner (2010): F totally real, Sf (E,K) 6= ∅.I ?


Notation

Let E be an elliptic curve over a number field F .I If v is an infinite real place of F , 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.Can compute only n ≤ 1, although construction works in general.

S(E,K) =v | NE∞F : v not split in K

.

Sign of functional equation for L(E/K, s) should be (−1)#S(E,K).I All constructions assume that #S(E,K) is odd.I In this talk: assume that S(E,K) = ν, with ν an infinite place.


Darmon’s quadratic ATR points

Let E be defined over a real quadratic F (with h+F = 1).

Suppose that K/F is an ATR extension:I The 1st embedding v1 of F extends to one complex place of K.I The 2nd embedding v2 of F extends to two real places of K.

Suppose that p | NE =⇒ p is split in K ( =⇒ S(E,K) = v1).Let τ ∈ K \ F . One has StabΓ0(NE)(v1(τ)) = 〈γτ 〉.Given τ2 ∈ H, consider the geodesic path joining τ2 with τ ′2 = γττ2.

×H H

τ1τ2

τ ′2

γτ

Γ0(NE)

X0(NE)

Fact: τ1 × γτ in Z1(Γ0(NE)\(H×H),Z) is null-homologous.; ∆τ a 2-chain such that ∂∆τ = τ1 × γτ .

I ∆τ is well-defined up to H2(X0(NE),Z).Marc Masdeu Analytic construction of pointson modular elliptic curves May 26, 2015 15 / 23

Darmon’s quadratic ATR points (II)

DefinitionA Hilbert modular form (HMF) of level N is a holomorphic functionf : H×H→ C, such that

f(γ1z1, γ2z2) = (c1z1 + d1)2(c2z2 + d2)2f(z1, z2), γ ∈ Γ0(N).

Have also Fourier expansions

fE(z1, z2) =∑n>>0

an(fE)e2πi(n1z1/δ1+n2z2/δ2).

Theorem (Freitas–Le-Hung–Siksek)There is a HMF fE of level NE such that ap(fE) = ap(E) for all p.


Darmon’s quadratic ATR points (III)

Need to symmetrize the HMF fE to account for units of F .I Yields fE , which is no longer holomorphic.

Integration yields an element Jτ =∫∫

∆τfEdz1dz2 ∈ C.

Well-defined up to a lattice

L =∫∫

∆fEdz1dz2 : ∆ ∈ H2(X0(NE),Z).

Conjecture 1 (Oda)There is an isogeny β : C/L→ E(C).

Pτ = β(Jτ ) ∈ E(C).

Conjecture 2 (Darmon)1 The local point Pτ is global, and belongs to E(Hτ ).2 Pτ is nontorsion if and only if L′(E/K, 1) 6= 0.


Archimedean cubic Darmon points (I)

Let F be a cubic field of signature (1, 1) (and h+F = 1).

Let E/F be an elliptic curve, of conductor NE .Consider the arithmetic group

Γ0(NE) ⊂ SL2(OF ) ⊂ SL2(R)× SL2(C).

H3 = C× R>0 = hyperbolic 3-space, on which SL2(C) acts:(a bc d

)· (x, y) =

((ax+ b)(cx+ d) + acy2

|cx+ d|2 + |cy|2 ,y

|cx+ d|2 + |cy|2

).

Get an action of Γ0(NE) on the symmetric space H×H3.

R>0

C

×

H H3


Archimedean cubic Darmon points (II)

Assume that E is modular.E ; an automorphic form ωE with Fourier-Bessel expansion:

ωE(z, x, y) =∑

α∈δ−1OFα0>0

a(δα)(E)e2πi(α0z+α1x+α2x)yH (α1y) ·(−dx∧dzdy∧dzdx∧dz

)

H(t) =

(− i

2eiθK1(4πρ),K0(4πρ),

i

2e−iθK1(4πρ)

)t = ρeiθ.

I K0 and K1 are hyperbolic Bessel functions of the second kind:

K0(x) =

∫ ∞0

e−x cosh(t)dt, K1(x) =

∫ ∞0

e−x cosh(t) cosh(t)dt.

ωE descends to a harmonic 2-form on Γ0(NE)\ (H×H3).


Archimedean cubic Darmon points (III)

Let K be a totally complex quadratic extension of F .I Suppose that p | NE =⇒ p is split in K. ( =⇒ S(E,K) = v1).

Choose τ ∈ K \ F .R>0

C

×

H H3

τ1

γτ

τ2

τ ′2

Γ0(NE)

τ ; ∆τ ∈ C2(Γ0(NE),Z).

Jτ =

∫∆τ

ωE ∈ C.

Jτ ; Pτ via C/ΛE → E(C).Conjecture: Pτ is defined over a finite abelian extension of K.


Remarks

When n > 1 the cycles constructed are analogous, but of higherdimension.

I Need to develop computational homology of arithmetic groups.

When #S(E,K) > 1 the group Γ0(NE) is replaced with the(norm-one) units of a certain quaternion algebra over F .

I No computations have been done in this setting.

There is a p-adic counterpart to all these constructions, where therole of the place v1 is substituted with a (finite) prime.


Example (I)

Let F = Q(r) with r3 − r2 + 1 = 0.

F signature (1, 1) and discriminant −23.

Consider the elliptic curve E/F given by the equation:

E/F : y2 + (r − 1)xy +(r2 − r

)y = x3 +

(−r2 − 1

)x2 + r2x.

E has prime conductor NE =(r2 + 4

)of norm 89.

K = F (α), with α2 + (r + 1)α+ 2r2 − 3r + 3 = 0.

I K has class number 1, thus we expect the point to be defined over K.


Example (II)

Take τ = α. This gives

γτ =

(−4r − 3 −r2 + 2r + 3

−2r2 − 4r − 3 −r2 + 4r + 2

)Finding ∆τ with ∂∆τ = τ × γτ amounts to decomposing γτ into aproduct of elementary matrices.

I Effective version of congruence subgroup problem.

Jτ =∑i

∫ si

ri

∫ γi·O

OωE(z, x, y).

We obtain, summing over all ideals (α) of norm up to 400, 000:

Jτ = 0.1419670770183− 0.0550994633√−1 ∈ C/ΛE ; Pτ ∈ E(C).

Numerically (up to 32 decimal digits) we obtain:

Pτ?= 10×

(r − 1, α− r2 + 2r

)∈ E(K).


Thank you !

“The fun of the subject seems to me to be in the examples.

B. Gross, in a letter to B. Birch, 1982”Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/


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