Automorphic forms and Number Theory International Center, Goa - - PowerPoint PPT Presentation

Automorphic forms and Number Theory

International Center, Goa August 2010

Apology I will not talk about p-adic weak harmonic Maass forms, as I had advertised...

The Birch and Swinnerton-Dyer conjecture for Q-curves and Oda’s period relations ... Joint work in progress with Victor Rotger (Barcelona), Yu Zhao (Montreal)

Henri Darmon

The Birch and Swinnerton-Dyer conjecture

E = an elliptic curve over a number field F. L(E/F, s) = its Hasse-Weil L-series. Conjecture (Birch and Swinnerton-Dyer) L(E/F, s) has analytic continuation to all s ∈ C and

• rds=1 L(E/F, s) = rank(E(F))

The BSD conjecture for analytic rank ≤ 1

Assume F = Q. Then L(E, s) is known to have analytic continuation thanks to modularity. Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite, and rank(E(Q)) = ords=1 L(E, s). Three key ingredients:

1 Modularity (in a strong geometric form); 2 Heegner points on modular curves and the Gross-Zagier

theorem;

3 Kolyvagin’s descent.

The BSD conjecture for analytic rank ≤ 1

Assume F = Q. Then L(E, s) is known to have analytic continuation thanks to modularity. Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite, and rank(E(Q)) = ords=1 L(E, s). Three key ingredients:

1 Modularity (in a strong geometric form); 2 Heegner points on modular curves and the Gross-Zagier

theorem;

3 Kolyvagin’s descent.

The BSD conjecture for analytic rank ≤ 1

Assume F = Q. Then L(E, s) is known to have analytic continuation thanks to modularity. Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite, and rank(E(Q)) = ords=1 L(E, s). Three key ingredients:

1 Modularity (in a strong geometric form); 2 Heegner points on modular curves and the Gross-Zagier

theorem;

3 Kolyvagin’s descent.

Modularity

Theorem (Geometric modularity) There is a non-constant morphism πE : J0(N) − → E, were J0(N) is the Jacobian of X0(N). The proof uses:

1 The modularity theorem (Wiles, Taylor-Wiles,

Breuil-Conrad-Diamond-Taylor);

2 The Tate conjecture for curves and abelian varieties over

number fields (Serre, Faltings).

Modularity

Theorem (Geometric modularity) There is a non-constant morphism πE : J0(N) − → E, were J0(N) is the Jacobian of X0(N). The proof uses:

1 The modularity theorem (Wiles, Taylor-Wiles,

Breuil-Conrad-Diamond-Taylor);

2 The Tate conjecture for curves and abelian varieties over

number fields (Serre, Faltings).

Heegner points

K= imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of OK of norm N, with OK/N ≃ Z/NZ. Definition The Heegner points on X0(N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + cOK. They are defined over the ring class field of K of conductor c. PK := πE((A1, A1[N]) + · · · + (Ah, Ah[N]) − h(∞)) ∈ E(K).

Heegner points

K= imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of OK of norm N, with OK/N ≃ Z/NZ. Definition The Heegner points on X0(N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + cOK. They are defined over the ring class field of K of conductor c. PK := πE((A1, A1[N]) + · · · + (Ah, Ah[N]) − h(∞)) ∈ E(K).

Heegner points

K= imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of OK of norm N, with OK/N ≃ Z/NZ. Definition The Heegner points on X0(N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + cOK. They are defined over the ring class field of K of conductor c. PK := πE((A1, A1[N]) + · · · + (Ah, Ah[N]) − h(∞)) ∈ E(K).

The Gross-Zagier Theorem

Theorem (Gross-Zagier) For all K satisfying (HH), the L-series L(E/K, s) vanishes to odd

• rder at s = 1, and

L′(E/K, 1) = PK, PKf , f (mod Q×). In particular, PK is of infinite order iff L′(E/K, 1) = 0.

Kolyvagin’s Theorem

Theorem (Kolyvagin) If PK is of infinite order, then rank(E(K)) = 1, and L L I(E/K) < ∞. The Heegner point PK is part of a norm-coherent system of algebraic points on E; This collection of points satisfies the axioms of an Euler system (a Kolyvagin system in the sense of Mazur-Rubin) which can be used to bound the p-Selmer group of E/K.

Kolyvagin’s Theorem

Theorem (Kolyvagin) If PK is of infinite order, then rank(E(K)) = 1, and L L I(E/K) < ∞. The Heegner point PK is part of a norm-coherent system of algebraic points on E; This collection of points satisfies the axioms of an Euler system (a Kolyvagin system in the sense of Mazur-Rubin) which can be used to bound the p-Selmer group of E/K.

Proof of BSD in analytic rank ≤ 1

Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite and rank(E(Q)) = ords=1 L(E, s). Proof.

• 1. Bump-Friedberg-Hoffstein, Murty-Murty ⇒ there exists a K

satisfying (HH), with ords=1 L(E/K, s) = 1.

• 2. Gross-Zagier ⇒ the Heegner point PK is of infinite order.
• 3. Koyvagin ⇒ E(K) ⊗ Q = Q · PK, and L

L I(E/K) < ∞.

• 4. Explicit calculation ⇒

. the point PK belongs to      E(Q) if L(E, 1) = 0, E(K)− if L(E, 1) = 0.

Proof of BSD in analytic rank ≤ 1

Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite and rank(E(Q)) = ords=1 L(E, s). Proof.

• 1. Bump-Friedberg-Hoffstein, Murty-Murty ⇒ there exists a K

satisfying (HH), with ords=1 L(E/K, s) = 1.

• 2. Gross-Zagier ⇒ the Heegner point PK is of infinite order.
• 3. Koyvagin ⇒ E(K) ⊗ Q = Q · PK, and L

L I(E/K) < ∞.

• 4. Explicit calculation ⇒

. the point PK belongs to      E(Q) if L(E, 1) = 0, E(K)− if L(E, 1) = 0.

Proof of BSD in analytic rank ≤ 1

Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite and rank(E(Q)) = ords=1 L(E, s). Proof.

• 1. Bump-Friedberg-Hoffstein, Murty-Murty ⇒ there exists a K

satisfying (HH), with ords=1 L(E/K, s) = 1.

• 2. Gross-Zagier ⇒ the Heegner point PK is of infinite order.
• 3. Koyvagin ⇒ E(K) ⊗ Q = Q · PK, and L

L I(E/K) < ∞.

• 4. Explicit calculation ⇒

. the point PK belongs to      E(Q) if L(E, 1) = 0, E(K)− if L(E, 1) = 0.

Proof of BSD in analytic rank ≤ 1

Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite and rank(E(Q)) = ords=1 L(E, s). Proof.

• 1. Bump-Friedberg-Hoffstein, Murty-Murty ⇒ there exists a K

satisfying (HH), with ords=1 L(E/K, s) = 1.

• 2. Gross-Zagier ⇒ the Heegner point PK is of infinite order.
• 3. Koyvagin ⇒ E(K) ⊗ Q = Q · PK, and L

L I(E/K) < ∞.

• 4. Explicit calculation ⇒

. the point PK belongs to      E(Q) if L(E, 1) = 0, E(K)− if L(E, 1) = 0.

Proof of BSD in analytic rank ≤ 1

Theorem (Gross-Zagier, Kolyvagin) If ords=1 L(E, s) ≤ 1, then L L I(E/Q) is finite and rank(E(Q)) = ords=1 L(E, s). Proof.

• 1. Bump-Friedberg-Hoffstein, Murty-Murty ⇒ there exists a K

satisfying (HH), with ords=1 L(E/K, s) = 1.

• 2. Gross-Zagier ⇒ the Heegner point PK is of infinite order.
• 3. Koyvagin ⇒ E(K) ⊗ Q = Q · PK, and L

L I(E/K) < ∞.

• 4. Explicit calculation ⇒

. the point PK belongs to      E(Q) if L(E, 1) = 0, E(K)− if L(E, 1) = 0.

Totally real fields

The mathematical objects exploited by Gross-Zagier and Kolyvagin continue to be available when Q is replaced by a totally real field F

• f degree n > 1.

Definition An elliptic curve E/F is modular if there is an automorphic representation π(E) of GL2(AF) attached to E, or, equivalently, a Hilbert modular form G ∈ S2(N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

Totally real fields

The mathematical objects exploited by Gross-Zagier and Kolyvagin continue to be available when Q is replaced by a totally real field F

• f degree n > 1.

Definition An elliptic curve E/F is modular if there is an automorphic representation π(E) of GL2(AF) attached to E, or, equivalently, a Hilbert modular form G ∈ S2(N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

Totally real fields

The mathematical objects exploited by Gross-Zagier and Kolyvagin continue to be available when Q is replaced by a totally real field F

• f degree n > 1.

Definition An elliptic curve E/F is modular if there is an automorphic representation π(E) of GL2(AF) attached to E, or, equivalently, a Hilbert modular form G ∈ S2(N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

Geometric modularity

Geometrically, the Hilbert modular form G corresponds to a (2n-dimensional) subspace ΩG ⊂ Ωn

har(V (C))G,

where V is a suitable Hilbert modular variety of dimension n. Definition The elliptic curve E/F is said to satisfy the Jacquet-Langlands hypothesis (JL) if either [F : Q] is odd, or there is at least one prime v|N at which πv(E) is not in the principal series. Theorem (Geometric modularity) Suppose that E/F is modular and satisfies (JL). There there exists a Shimura curve X/F and a non-constant morphism πE : Jac(X) − → E.

Geometric modularity

Geometrically, the Hilbert modular form G corresponds to a (2n-dimensional) subspace ΩG ⊂ Ωn

har(V (C))G,

where V is a suitable Hilbert modular variety of dimension n. Definition The elliptic curve E/F is said to satisfy the Jacquet-Langlands hypothesis (JL) if either [F : Q] is odd, or there is at least one prime v|N at which πv(E) is not in the principal series. Theorem (Geometric modularity) Suppose that E/F is modular and satisfies (JL). There there exists a Shimura curve X/F and a non-constant morphism πE : Jac(X) − → E.

Geometric modularity

Geometrically, the Hilbert modular form G corresponds to a (2n-dimensional) subspace ΩG ⊂ Ωn

har(V (C))G,

where V is a suitable Hilbert modular variety of dimension n. Definition The elliptic curve E/F is said to satisfy the Jacquet-Langlands hypothesis (JL) if either [F : Q] is odd, or there is at least one prime v|N at which πv(E) is not in the principal series. Theorem (Geometric modularity) Suppose that E/F is modular and satisfies (JL). There there exists a Shimura curve X/F and a non-constant morphism πE : Jac(X) − → E.

Zhang’s Theorem

Shimura curves, like modular curves, are equipped with a plentiful supply of CM points. Theorem (Zhang) Let E/F be a modular elliptic curve satisfying hypothesis (JL). If

• rds=1 L(E/F, s) ≤ 1, then L

L I(E/F) is finite and rank(E(F)) = ords=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves.

• Ann. of Math. (2) 153 (2001).

Zhang’s Theorem

Shimura curves, like modular curves, are equipped with a plentiful supply of CM points. Theorem (Zhang) Let E/F be a modular elliptic curve satisfying hypothesis (JL). If

• rds=1 L(E/F, s) ≤ 1, then L

L I(E/F) is finite and rank(E(F)) = ords=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves.

• Ann. of Math. (2) 153 (2001).

Zhang’s Theorem

Shimura curves, like modular curves, are equipped with a plentiful supply of CM points. Theorem (Zhang) Let E/F be a modular elliptic curve satisfying hypothesis (JL). If

• rds=1 L(E/F, s) ≤ 1, then L

L I(E/F) is finite and rank(E(F)) = ords=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves.

• Ann. of Math. (2) 153 (2001).

BSD in analytic rank zero

In analytic rank zero one can dispense with (JL). Theorem (Matteo Longo) Let E/F be a modular elliptic curve. If L(E/F, 1) = 0, then E(F) is finite and L L I(E/F)[p∞] is finite for almost all p. Proof. Congruences between modular forms ⇒ the Galois representation E[pn] occurs in Jn[pn], where Jn = Jac(Xn) and Xn is a Shimura curve Xn whose level may (and does) depend on n. Use CM points on Xn to bound the pn-Selmer group of E.

BSD in analytic rank zero

In analytic rank zero one can dispense with (JL). Theorem (Matteo Longo) Let E/F be a modular elliptic curve. If L(E/F, 1) = 0, then E(F) is finite and L L I(E/F)[p∞] is finite for almost all p. Proof. Congruences between modular forms ⇒ the Galois representation E[pn] occurs in Jn[pn], where Jn = Jac(Xn) and Xn is a Shimura curve Xn whose level may (and does) depend on n. Use CM points on Xn to bound the pn-Selmer group of E.

BSD in analytic rank zero

In analytic rank zero one can dispense with (JL). Theorem (Matteo Longo) Let E/F be a modular elliptic curve. If L(E/F, 1) = 0, then E(F) is finite and L L I(E/F)[p∞] is finite for almost all p. Proof. Congruences between modular forms ⇒ the Galois representation E[pn] occurs in Jn[pn], where Jn = Jac(Xn) and Xn is a Shimura curve Xn whose level may (and does) depend on n. Use CM points on Xn to bound the pn-Selmer group of E.

The Challenge that remains

When ords=1 L(E/F, s) = 1 but hypothesis (JL) is not satisfied, produce the point in E(F) whose existence is predicted by BSD. Remark: If E/F does not satisfy (JL), then its conductor is a square. Prototypical case where (JL) fails to hold: F = Q( √ N), a real quadratic field, cond(E/F) = 1. I will focus on this case for simplicity. Fact: E(F) has even analytic rank, so Longo’s theorem applies.

The Challenge that remains

When ords=1 L(E/F, s) = 1 but hypothesis (JL) is not satisfied, produce the point in E(F) whose existence is predicted by BSD. Remark: If E/F does not satisfy (JL), then its conductor is a square. Prototypical case where (JL) fails to hold: F = Q( √ N), a real quadratic field, cond(E/F) = 1. I will focus on this case for simplicity. Fact: E(F) has even analytic rank, so Longo’s theorem applies.

The Challenge that remains

When ords=1 L(E/F, s) = 1 but hypothesis (JL) is not satisfied, produce the point in E(F) whose existence is predicted by BSD. Remark: If E/F does not satisfy (JL), then its conductor is a square. Prototypical case where (JL) fails to hold: F = Q( √ N), a real quadratic field, cond(E/F) = 1. I will focus on this case for simplicity. Fact: E(F) has even analytic rank, so Longo’s theorem applies.

The Challenge that remains

When ords=1 L(E/F, s) = 1 but hypothesis (JL) is not satisfied, produce the point in E(F) whose existence is predicted by BSD. Remark: If E/F does not satisfy (JL), then its conductor is a square. Prototypical case where (JL) fails to hold: F = Q( √ N), a real quadratic field, cond(E/F) = 1. I will focus on this case for simplicity. Fact: E(F) has even analytic rank, so Longo’s theorem applies.

ATR twists

Consider the twist EM of E by a quadratic extension M/F. Proposition

1 If M is totally real or CM, then EM has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then EM has

• dd analytic rank.

Conjecture (on ATR twists) Let M be an ATR extension of F and let EM be the associated twist of E. If L′(EM/F, 1) = 0, then EM(F) has rank one and L L I(EM/F) < ∞. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

ATR twists

Consider the twist EM of E by a quadratic extension M/F. Proposition

1 If M is totally real or CM, then EM has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then EM has

• dd analytic rank.

Conjecture (on ATR twists) Let M be an ATR extension of F and let EM be the associated twist of E. If L′(EM/F, 1) = 0, then EM(F) has rank one and L L I(EM/F) < ∞. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

ATR twists

Consider the twist EM of E by a quadratic extension M/F. Proposition

1 If M is totally real or CM, then EM has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then EM has

• dd analytic rank.

Conjecture (on ATR twists) Let M be an ATR extension of F and let EM be the associated twist of E. If L′(EM/F, 1) = 0, then EM(F) has rank one and L L I(EM/F) < ∞. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

ATR twists

Consider the twist EM of E by a quadratic extension M/F. Proposition

1 If M is totally real or CM, then EM has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then EM has

• dd analytic rank.

Conjecture (on ATR twists) Let M be an ATR extension of F and let EM be the associated twist of E. If L′(EM/F, 1) = 0, then EM(F) has rank one and L L I(EM/F) < ∞. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

ATR cycles

Some years ago, Adam Logan and I proposed a strategy for calculating a global point on EM(F), based on Abel-Jacobi images

• f ATR cycles.

Let Y be the (open) Hilbert modular surface attached to E/F: Y (C) = SL2(OF)\(H1 × H2). To any OF-algebra embedding Ψ : OM − → M2(OF),

• ne can attach cycles ∆ψ ⊂ Y (C) of real dimension one which

“behave like Heegner points”.

ATR cycles

Some years ago, Adam Logan and I proposed a strategy for calculating a global point on EM(F), based on Abel-Jacobi images

• f ATR cycles.

Let Y be the (open) Hilbert modular surface attached to E/F: Y (C) = SL2(OF)\(H1 × H2). To any OF-algebra embedding Ψ : OM − → M2(OF),

• ne can attach cycles ∆ψ ⊂ Y (C) of real dimension one which

“behave like Heegner points”.

ATR cycles

Some years ago, Adam Logan and I proposed a strategy for calculating a global point on EM(F), based on Abel-Jacobi images

• f ATR cycles.

Let Y be the (open) Hilbert modular surface attached to E/F: Y (C) = SL2(OF)\(H1 × H2). To any OF-algebra embedding Ψ : OM − → M2(OF),

• ne can attach cycles ∆ψ ⊂ Y (C) of real dimension one which

“behave like Heegner points”.

ATR cycles

τ (1)

Ψ

:= fixed point of Ψ(M×) H1; τ (2)

Ψ , τ (2)′ Ψ

:= fixed points of Ψ(M×) (H2 ∪ R); ΥΨ = {τ (1)

Ψ } × geodesic(τ (2) Ψ

→ τ (2)′

Ψ ).

• τ(1)

Ψ

×

τ (2)

Ψ

τ (2)′

Ψ

• ∆Ψ = ΥΨ/Ψ(O×

M) ⊂ Y (C).

Key fact: The cycles ∆Ψ are null-homologous.

ATR cycles

τ (1)

Ψ

:= fixed point of Ψ(M×) H1; τ (2)

Ψ , τ (2)′ Ψ

:= fixed points of Ψ(M×) (H2 ∪ R); ΥΨ = {τ (1)

Ψ } × geodesic(τ (2) Ψ

→ τ (2)′

Ψ ).

• τ(1)

Ψ

×

τ (2)

Ψ

τ (2)′

Ψ

• ∆Ψ = ΥΨ/Ψ(O×

M) ⊂ Y (C).

Key fact: The cycles ∆Ψ are null-homologous.

ATR cycles

τ (1)

Ψ

:= fixed point of Ψ(M×) H1; τ (2)

Ψ , τ (2)′ Ψ

:= fixed points of Ψ(M×) (H2 ∪ R); ΥΨ = {τ (1)

Ψ } × geodesic(τ (2) Ψ

→ τ (2)′

Ψ ).

• τ(1)

Ψ

×

τ (2)

Ψ

τ (2)′

Ψ

• ∆Ψ = ΥΨ/Ψ(O×

M) ⊂ Y (C).

Key fact: The cycles ∆Ψ are null-homologous.

ATR cycles

τ (1)

Ψ

:= fixed point of Ψ(M×) H1; τ (2)

Ψ , τ (2)′ Ψ

:= fixed points of Ψ(M×) (H2 ∪ R); ΥΨ = {τ (1)

Ψ } × geodesic(τ (2) Ψ

→ τ (2)′

Ψ ).

• τ(1)

Ψ

×

τ (2)

Ψ

τ (2)′

Ψ

• ∆Ψ = ΥΨ/Ψ(O×

M) ⊂ Y (C).

Key fact: The cycles ∆Ψ are null-homologous.

ATR cycles

τ (1)

Ψ

:= fixed point of Ψ(M×) H1; τ (2)

Ψ , τ (2)′ Ψ

:= fixed points of Ψ(M×) (H2 ∪ R); ΥΨ = {τ (1)

Ψ } × geodesic(τ (2) Ψ

→ τ (2)′

Ψ ).

• τ(1)

Ψ

×

τ (2)

Ψ

τ (2)′

Ψ

• ∆Ψ = ΥΨ/Ψ(O×

M) ⊂ Y (C).

Key fact: The cycles ∆Ψ are null-homologous.

Oda’s periods

ATR cycles are similar to the modular symbols on Hilbert modular varieties of Mladen Dimitrov’s lecture, whose classes in homology encode special values of L-functions. Since ATR cycles are null-homologous, one may hope to relate them to first derivatives. For any 2-form ωG ∈ ΩG, P?

Ψ(G) :=

• ∂−1∆Ψ

ωG ∈ C/ΛG Conjecture (Oda) For a suitable choice of ωG, we have C/ΛG ∼ E(C). In particular P?

Ψ(G) can then be viewed as a point in E(C).

Oda’s periods

ATR cycles are similar to the modular symbols on Hilbert modular varieties of Mladen Dimitrov’s lecture, whose classes in homology encode special values of L-functions. Since ATR cycles are null-homologous, one may hope to relate them to first derivatives. For any 2-form ωG ∈ ΩG, P?

Ψ(G) :=

• ∂−1∆Ψ

ωG ∈ C/ΛG Conjecture (Oda) For a suitable choice of ωG, we have C/ΛG ∼ E(C). In particular P?

Ψ(G) can then be viewed as a point in E(C).

Oda’s periods

ATR cycles are similar to the modular symbols on Hilbert modular varieties of Mladen Dimitrov’s lecture, whose classes in homology encode special values of L-functions. Since ATR cycles are null-homologous, one may hope to relate them to first derivatives. For any 2-form ωG ∈ ΩG, P?

Ψ(G) :=

• ∂−1∆Ψ

ωG ∈ C/ΛG Conjecture (Oda) For a suitable choice of ωG, we have C/ΛG ∼ E(C). In particular P?

Ψ(G) can then be viewed as a point in E(C).

Oda’s periods

ATR cycles are similar to the modular symbols on Hilbert modular varieties of Mladen Dimitrov’s lecture, whose classes in homology encode special values of L-functions. Since ATR cycles are null-homologous, one may hope to relate them to first derivatives. For any 2-form ωG ∈ ΩG, P?

Ψ(G) :=

• ∂−1∆Ψ

ωG ∈ C/ΛG Conjecture (Oda) For a suitable choice of ωG, we have C/ΛG ∼ E(C). In particular P?

Ψ(G) can then be viewed as a point in E(C).

ATR points

Conjecture (Logan, D, 2004) The points P?

Ψ(G) belong to E(H) ⊗ Q, where H is the Hilbert

class field of M. The points P?

Ψ1(G), . . . , P? Ψh(G) are conjugate to

each other under Gal(H/M). Finally, the point P?

M(G) := P? Ψ1(G) + · · · + P? Ψh(G)

is of infinite order iff L′(E/M, 1) = 0. This conjecture (in a sufficiently general and precise form) would imply the Conjecture on ATR twists. But we do not know how to tackle it.

ATR points

Conjecture (Logan, D, 2004) The points P?

Ψ(G) belong to E(H) ⊗ Q, where H is the Hilbert

class field of M. The points P?

Ψ1(G), . . . , P? Ψh(G) are conjugate to

each other under Gal(H/M). Finally, the point P?

M(G) := P? Ψ1(G) + · · · + P? Ψh(G)

is of infinite order iff L′(E/M, 1) = 0. This conjecture (in a sufficiently general and precise form) would imply the Conjecture on ATR twists. But we do not know how to tackle it.

ATR points

Conjecture (Logan, D, 2004) The points P?

Ψ(G) belong to E(H) ⊗ Q, where H is the Hilbert

class field of M. The points P?

Ψ1(G), . . . , P? Ψh(G) are conjugate to

each other under Gal(H/M). Finally, the point P?

M(G) := P? Ψ1(G) + · · · + P? Ψh(G)

is of infinite order iff L′(E/M, 1) = 0. This conjecture (in a sufficiently general and precise form) would imply the Conjecture on ATR twists. But we do not know how to tackle it.

ATR points

Conjecture (Logan, D, 2004) The points P?

Ψ(G) belong to E(H) ⊗ Q, where H is the Hilbert

class field of M. The points P?

Ψ1(G), . . . , P? Ψh(G) are conjugate to

each other under Gal(H/M). Finally, the point P?

M(G) := P? Ψ1(G) + · · · + P? Ψh(G)

is of infinite order iff L′(E/M, 1) = 0. This conjecture (in a sufficiently general and precise form) would imply the Conjecture on ATR twists. But we do not know how to tackle it.

The current work with Rotger and Zhao: Q-curves

Definition A Q-curve over F is an elliptic curve E/F which is F-isogenous to its Galois conjugate. Theorem (Ribet) Let E be a Q-curve of conductor 1 over F = Q( √ N). Then there is an elliptic modular form f ∈ S2(Γ1(N), εF) with fourier coefficients in a quadratic (imaginary) field such that L(E/F, s) = L(f , s)L(f σ, s). The Hilbert modular form G on GL2(AF) is the Doi-Naganuma lift

• f f . Modular parametrisation defined over F:

J1(N) − → E.

The current work with Rotger and Zhao: Q-curves

Definition A Q-curve over F is an elliptic curve E/F which is F-isogenous to its Galois conjugate. Theorem (Ribet) Let E be a Q-curve of conductor 1 over F = Q( √ N). Then there is an elliptic modular form f ∈ S2(Γ1(N), εF) with fourier coefficients in a quadratic (imaginary) field such that L(E/F, s) = L(f , s)L(f σ, s). The Hilbert modular form G on GL2(AF) is the Doi-Naganuma lift

• f f . Modular parametrisation defined over F:

J1(N) − → E.

The current work with Rotger and Zhao: Q-curves

Definition A Q-curve over F is an elliptic curve E/F which is F-isogenous to its Galois conjugate. Theorem (Ribet) Let E be a Q-curve of conductor 1 over F = Q( √ N). Then there is an elliptic modular form f ∈ S2(Γ1(N), εF) with fourier coefficients in a quadratic (imaginary) field such that L(E/F, s) = L(f , s)L(f σ, s). The Hilbert modular form G on GL2(AF) is the Doi-Naganuma lift

• f f . Modular parametrisation defined over F:

J1(N) − → E.

The current work with Rotger and Zhao: Q-curves

Definition A Q-curve over F is an elliptic curve E/F which is F-isogenous to its Galois conjugate. Theorem (Ribet) Let E be a Q-curve of conductor 1 over F = Q( √ N). Then there is an elliptic modular form f ∈ S2(Γ1(N), εF) with fourier coefficients in a quadratic (imaginary) field such that L(E/F, s) = L(f , s)L(f σ, s). The Hilbert modular form G on GL2(AF) is the Doi-Naganuma lift

• f f . Modular parametrisation defined over F:

J1(N) − → E.

The Theorem on ATR twists for Q-curves

Recall: If E is a Q-curve, then E/F has even analytic rank; the same is true for its twists by CM or totally real quadratic characters χ of F with χ(N) = 1. Theorem (Victor Rotger, Yu Zhao, D) Let E be a Q-curve of square conductor NE over a real quadratic field F, and let M/F be an ATR extension of discriminant prime to

• NE. If L′(EM/F, 1) = 0, then EM(F) has rank one and L

L I(EM/F) is finite.

The Theorem on ATR twists for Q-curves

Recall: If E is a Q-curve, then E/F has even analytic rank; the same is true for its twists by CM or totally real quadratic characters χ of F with χ(N) = 1. Theorem (Victor Rotger, Yu Zhao, D) Let E be a Q-curve of square conductor NE over a real quadratic field F, and let M/F be an ATR extension of discriminant prime to

• NE. If L′(EM/F, 1) = 0, then EM(F) has rank one and L

L I(EM/F) is finite.

Elliptic curves of conductor 1

Pinch, Cremona: For N = disc(F) prime and ≤ 1000, there are exactly 17 isogeny classes of elliptic curves of conductor 1 over Q( √ N), N = 29, 37, 41, 109, 157, 229, 257, 337, 349, 397, 461, 509, 509, 877, 733, 881, 997. All but two (N = 509, 877) are Q-curves.

Some Galois theory

Let M= Galois closure of M over Q. Then Gal(M/Q) = D8. This group contains two copies of the Klein 4-group: VF = τM, τ ′

M,

VK = τL, τ ′

L. τ ′

M

• τL
• τM
• τ ′

L

Some Galois theory

Suppose that F = MVF M = MτM M′ = Mτ ′

M,

and set K = MVK L = MτL L′ = Mτ ′

L.

M M

• M′
• Q(

√ N, √ −d) L′

• L
• F
• Q(

√ −Nd) K

• Q

Key facts about K and L

Let      χM : A×

F −

→ ±1 be the quadratic character attached to M/F; χL : A×

K −

→ ±1 be the quadratic character attached to L/K.

1 K = Q(

√ −d) is an imaginary quadratic field, and satisfies (HH);

2 The central character χL|A× Q is equal to εF. 3 IndQ

F χM = IndQ K χL;

Key facts about K and L

Let      χM : A×

F −

→ ±1 be the quadratic character attached to M/F; χL : A×

K −

→ ±1 be the quadratic character attached to L/K.

1 K = Q(

√ −d) is an imaginary quadratic field, and satisfies (HH);

2 The central character χL|A× Q is equal to εF. 3 IndQ

F χM = IndQ K χL;

Key facts about K and L

Let      χM : A×

F −

→ ±1 be the quadratic character attached to M/F; χL : A×

K −

→ ±1 be the quadratic character attached to L/K.

1 K = Q(

√ −d) is an imaginary quadratic field, and satisfies (HH);

2 The central character χL|A× Q is equal to εF. 3 IndQ

F χM = IndQ K χL;

Key facts about K and L

Let      χM : A×

F −

→ ±1 be the quadratic character attached to M/F; χL : A×

K −

→ ±1 be the quadratic character attached to L/K.

1 K = Q(

√ −d) is an imaginary quadratic field, and satisfies (HH);

2 The central character χL|A× Q is equal to εF. 3 IndQ

F χM = IndQ K χL;

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

The Artin formalism

Let E/F be a Q-curve and let f ∈ S2(Γ0(N), εF) be the associated elliptic cusp form. L(EM/F, s) = L(E/F, χM, s) = L(f /F, χM, s) = L(f ⊗ IndQ

F χM, s)

= L(f ⊗ IndQ

K χL, s) = L(f /K, χL, s).

In particular, L′(EM/F, 1) = 0 implies that L′(f /K, χL, 1) = 0.

A theorem of GZK-type

The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ1(N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L′(f /K, χL, 1) = 0, then Af (L)− ⊗ Q has dimension one over Tf , and therefore rank(Af (L)−) = 2. Furthermore L L I(Af /L)− is finite. rank(Af (L)−) = rank(Af (M)−), Af (M)− = E(M)− ⊕ E(M)−. Corollary If L′(EM/F, 1) = 0, then rank(EM(F)) = 1 and L L I(EM/F) < ∞.

A theorem of GZK-type

The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ1(N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L′(f /K, χL, 1) = 0, then Af (L)− ⊗ Q has dimension one over Tf , and therefore rank(Af (L)−) = 2. Furthermore L L I(Af /L)− is finite. rank(Af (L)−) = rank(Af (M)−), Af (M)− = E(M)− ⊕ E(M)−. Corollary If L′(EM/F, 1) = 0, then rank(EM(F)) = 1 and L L I(EM/F) < ∞.

A theorem of GZK-type

The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ1(N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L′(f /K, χL, 1) = 0, then Af (L)− ⊗ Q has dimension one over Tf , and therefore rank(Af (L)−) = 2. Furthermore L L I(Af /L)− is finite. rank(Af (L)−) = rank(Af (M)−), Af (M)− = E(M)− ⊕ E(M)−. Corollary If L′(EM/F, 1) = 0, then rank(EM(F)) = 1 and L L I(EM/F) < ∞.

A final question

Underlying this theorem is the construction of a “classical” Heegner point PM(f ) ∈ EM(F). Question Is there a direct formula relating the ATR point P?

M(G) and the

“classical” Heegner point PM(f ) arising from J1(N)? The study undertaken with Rotger and Zhao suggests a relation of the form P?

M(G) ?

= ℓ · PM(f ), ℓ ∈ Q×. This statement ressembles the period relations of Oda relating the periods of an elliptic cusp form with those of its Doi-Naganuma lift, and hence might be more tractable (both computationally, and theoretically) than my original conjecture with Logan.

A final question

Underlying this theorem is the construction of a “classical” Heegner point PM(f ) ∈ EM(F). Question Is there a direct formula relating the ATR point P?

M(G) and the

“classical” Heegner point PM(f ) arising from J1(N)? The study undertaken with Rotger and Zhao suggests a relation of the form P?

M(G) ?

= ℓ · PM(f ), ℓ ∈ Q×. This statement ressembles the period relations of Oda relating the periods of an elliptic cusp form with those of its Doi-Naganuma lift, and hence might be more tractable (both computationally, and theoretically) than my original conjecture with Logan.

A final question

Underlying this theorem is the construction of a “classical” Heegner point PM(f ) ∈ EM(F). Question Is there a direct formula relating the ATR point P?

M(G) and the

“classical” Heegner point PM(f ) arising from J1(N)? The study undertaken with Rotger and Zhao suggests a relation of the form P?

M(G) ?

= ℓ · PM(f ), ℓ ∈ Q×. This statement ressembles the period relations of Oda relating the periods of an elliptic cusp form with those of its Doi-Naganuma lift, and hence might be more tractable (both computationally, and theoretically) than my original conjecture with Logan.

A final question

Underlying this theorem is the construction of a “classical” Heegner point PM(f ) ∈ EM(F). Question Is there a direct formula relating the ATR point P?

M(G) and the

“classical” Heegner point PM(f ) arising from J1(N)? The study undertaken with Rotger and Zhao suggests a relation of the form P?

M(G) ?

= ℓ · PM(f ), ℓ ∈ Q×. This statement ressembles the period relations of Oda relating the periods of an elliptic cusp form with those of its Doi-Naganuma lift, and hence might be more tractable (both computationally, and theoretically) than my original conjecture with Logan.

My usual apology Sorry for running over time!

A Big Thank You to Pierre Colmez, Wee Teck Gan, Eknath Ghate, Dipendra Prasad, Kenneth Ribet,1 Vinayak Vatsal.1

for organising this inspiring conference in such a lovely setting!

1Even if he did not show up...