1. The document discusses analytic constructions of points on modular elliptic curves over number fields. 2. It introduces Heegner points, which provide a tool for verifying the Birch and Swinnerton-Dyer conjecture when the number field is an imaginary quadratic field. 3. Later work has generalized these constructions to some real quadratic fields and cubic fields of signature (1,1) by using Hilbert modular forms and automorphic forms on hyperbolic 3-space.

1. Analytic construction of points
on modular elliptic curves
Congreso de J´ovenes Investigadores
Universidad 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

2. 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 = rkZ E(K) is called the algebraic rank of E(K).
Open problem: Given E and K, find r.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 1 / 26

3. The Hasse-Weil L-function
Suppose that K = Q(
√
D) is quadratic.
Can reduce the coefficients a1, . . . , a6 modulo primes p.
For almost all primes, the reduction is an elliptic curve (nonsingular).
Obviously E(Fp) is finite.
The conductor of E is an integer N encoding the shape of E when
this reduction is singular.
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)
=⇒
Analytic continuation of L(E/K, s) to C.
Functional equation relating s ↔ 2 − s.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 2 / 26

4. The BSD conjecture
Bryan Birch Sir Peter Swinnerton-Dyer
BSD conjecture (coarse version)
ords=1 L(E/K, s) = rkZ E(K).
So L(E/K, 1) = 0
BSD
=⇒ ∃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

5. Modular forms
Let N > 0 be an integer and consider
Γ0(N) = { a b
c d ∈ SL2(Z) : N | c}.
Γ0(N) acts on the upper-half plane H = {z ∈ C : Im(z) > 0}:
Via a b
c 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)2
f(z) for all γ = a b
c 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.
A newform is a simultaneous eigenvector for the Hecke algebra. (. . . )
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 4 / 26

6. 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 by
adding 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

7. 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 6 / 26

8. 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:
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

9. Heegner Points (K/Q imaginary quadratic)
Modularity =⇒ ∃ modular form fE attached to E.
ωE = 2πifE(z)dz = 2πi
n≥1
an(f)e2πinz
dz ∈ Ω1
Γ0(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) = 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

10. 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

11. Generalization
One can replace Q with any totally real field F.
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.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.
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

12. 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

13. Darmon’s Idea
Henri Darmon
What if K/F is not CM?
Simplest case: F = Q, K real quadratic.
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

14. History
Henri Darmon Adam Logan Xevi Guitart J´erˆome G¨artner
H. Darmon (2000): F totally real.
Darmon-Logan (2003): F quadratic norm-euclidean, NE trivial.
Guitart-M. (2011): F quadratic norm-euclidean, NE trivial.
Guitart-M. (2012): F quadratic norm-euclidean, NE trivial.
J. Gartner (2010): F totally real, relaxed Heegner hypothesis.
?
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 13 / 26

15. Notation
Consider a number field F.
If v is an infinite real place of F, then:
1 It may extend to two real places of K (splits), or
2 It may extend to one complex place of K (ramifies).
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).
All constructions assume that #S(E, K) is odd.
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

16. 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

17. Darmon’s quadratic ATR points
Let E be defined over a real quadratic F (with h+
F = 1).
Definition
A Hilbert modular form (HMF) of level N is a holomorphic function
f : H × H → C, such that
f(γ1z1, γ2z2) = (c1z1 + d1)2
(c2z2 + d2)2
f(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

18. Darmon’s quadratic ATR points (II)
Suppose that K/F is an ATR extension:
The 1st
embedding v1 of F extends to one complex place of K.
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 τ2 = γτ τ2.
×
H H
τ1
τ2
τ2
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 .
∆τ is well-defined up to H2(X0(NE), Z).
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 17 / 26

19. Darmon’s quadratic ATR points (III)
Need to symmetrize the HMF fE to account for units of F.
Yields ˜fE, which is no longer holomorphic.
Integration yields an element Jτ = ∆τ
˜fEdz1dz2 ∈ 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) = 0.
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 18 / 26

20. 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

21. 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 b
c d
· (x, y) =
(ax + b)(cx + d) + a¯cy2
|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

22. Cubic Darmon points (II)
Assume that E is modular.
E ; an automorphic form ωE with Fourier-Bessel expansion:
ωE(z, x, y) =
α∈δ−1OF
α0>0
a(δα)(E)e2πi(α0z+α1x+α2 ¯x)
yH (α1y) ·
−dx∧dz
dy∧dz
d¯x∧dz
H(t) = −
i
2
eiθ
K1(4πρ), K0(4πρ),
i
2
e−iθ
K1(4πρ) t = ρeiθ
.
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

23. Cubic Darmon points (III)
Let K be a totally complex quadratic extension of F.
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
τ2
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

24. Remarks
Conjectures say the exact extension Hτ /K over which Pτ is defined.
When n > 1 construct analogous cycles, but of higher dimension.
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.
No computations have been done in this setting.
There is a p-adic counterpart to all these constructions, where the
role of the place v1 is substituted with a (finite) prime.
See tomorrow’s talk by Carlos de Vera Piquero!
Marc Masdeu Analytic construction of pointson modular elliptic curves September 10, 2015 23 / 26

25. 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

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
+ r2
x.
E has prime conductor NE = r2 + 4 of norm 89.
K = F(α), with α2 + (r + 1) α + 2r2 − 3r + 3 = 0.
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

27. 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 a
product of elementary matrices.
Effective version of congruence subgroup problem.
Jτ =
i
si
ri
γi·O
O
ωE(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

28. 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