analytic construction of points on modular elliptic curves
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Analytic construction of pointson modular elliptic curves
Congreso de Jovenes InvestigadoresUniversidad de Murcia
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Universitat de Barcelona
2University of Warwick
3University of Sheffield
September 10, 2015
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 0 / 26
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 1 / 26
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 2 / 26
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 September 10, 2015 3 / 26
Modular forms
Let N > 0 be an integer and consider
Γ0(N) = (a bc d
)∈ SL2(Z) : N | c.
Γ0(N) acts on the upper-half plane H = z ∈ C : Im(z) > 0:I 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. (. . . )
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 4 / 26
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 5 / 26
Plan
1 Heegner points
2 After Heegner
3 Quadratic ATR points
4 Cubic (1, 1) points
5 Example
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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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 7 / 26
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.
Given τ ∈ K ∩H, set Jτ =
∫ τ
i∞ωE ∈ C.
Well-defined up to ΛE =∫
γ ωE | γ ∈ H1 (X0(N),Z)
.
Theorem (Weierstrass Uniformization)There exists a computable complex-analytic group isomorphism
η : C/ΛE → E(C), ΛE = lattice of rank 2.
Theorem (Shimura, Gross–Zagier, Kolyvagin)1 Pτ = η(Jτ ) ∈ E(Kab) ⊂ E(C).2 PK = Tr(Pτ ) is nontorsion ⇐⇒ L′(E/K, 1) 6= 0.3 If ords=1 L(E/Q, s) ≤ 1 then BSD holds for E(Q).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 8 / 26
Heegner Points: why did this work?
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 is defined on X0(N)(Kab).An explicit description of φ shows that:
Pτ = φE(τ) ∈ E(Kab).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 9 / 26
Generalization
One can replace Q with any totally real field F .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 .The Heegner hypothesis can be relaxed to:
Heegner Hypothesis: [F : Q] + #p | NE : p inert in K is odd.
I This still ensures that ords=1 L(E/K, s) is odd.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 10 / 26
Plan
1 Heegner points
2 After Heegner
3 Quadratic ATR points
4 Cubic (1, 1) points
5 Example
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 11 / 26
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?
. . . this may get us in trouble!
1 Algebraic model X/F .
2 Geometric modularity: φE : X → E.
3 CM points τ ∈ X(Kab).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 12 / 26
History
Henri Darmon Adam Logan Xevi Guitart Jerome Gartner
H. Darmon (2000): F totally real.
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, relaxed Heegner hypothesis.
I ?
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 13 / 26
Notation
Consider 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.
Let E be an elliptic curve over F .
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 14 / 26
Plan
1 Heegner points
2 After Heegner
3 Quadratic ATR points
4 Cubic (1, 1) points
5 Example
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 15 / 26
Darmon’s quadratic ATR points
Let E be defined over a real quadratic F (with h+F = 1).
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 16 / 26
Darmon’s quadratic ATR points (II)
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)(τ) = 〈γτ 〉. Set τ1 = v1(τ).Given τ2 ∈ H, consider the geodesic 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 September 10, 2015 17 / 26
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(Kab).2 Pτ is nontorsion if and only if L′(E/K, 1) 6= 0.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 18 / 26
Plan
1 Heegner points
2 After Heegner
3 Quadratic ATR points
4 Cubic (1, 1) points
5 Example
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 19 / 26
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
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 20 / 26
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).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 21 / 26
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 22 / 26
Remarks
Conjectures say the exact extension Hτ/K over which Pτ is defined.
When n > 1 construct analogous cycles, but of higher dimension.I Need to develop computational (co)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 , and X0(N)is replaced with a Shimura curve.
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.
I See tomorrow’s talk by Carlos de Vera Piquero!
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 23 / 26
Plan
1 Heegner points
2 After Heegner
3 Quadratic ATR points
4 Cubic (1, 1) points
5 Example
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 24 / 26
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.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 25 / 26
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).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 26 / 26
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/
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 26 / 26
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