Laval University, Québec September 2010

Transcription

1 Conférence Québec-Maine Laval University, Québec September 2010

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

3 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 ord s=1 L(E/F, s) = rank(e(f ))

4 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 ord s=1 L(E, s) 1, then LI (E/Q) is finite, and rank(e(q)) = ord s=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.

5 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 ord s=1 L(E, s) 1, then LI (E/Q) is finite, and rank(e(q)) = ord s=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.

6 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 ord s=1 L(E, s) 1, then LI (E/Q) is finite, and rank(e(q)) = ord s=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.

7 Modularity Theorem (Geometric modularity) There is a non-constant morphism π E : J 0 (N) E, were J 0 (N) is the Jacobian of X 0 (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).

8 Modularity Theorem (Geometric modularity) There is a non-constant morphism π E : J 0 (N) E, were J 0 (N) is the Jacobian of X 0 (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).

9 Heegner points K = imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of O K of norm N, with O K /N Z/NZ. Definition The Heegner points on X 0 (N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + co K. They are defined over the ring class field of K of conductor c. P K := π E ((A 1, A 1 [N]) + + (A h, A h [N]) h( )) E(K).

10 Heegner points K = imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of O K of norm N, with O K /N Z/NZ. Definition The Heegner points on X 0 (N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + co K. They are defined over the ring class field of K of conductor c. P K := π E ((A 1, A 1 [N]) + + (A h, A h [N]) h( )) E(K).

11 Heegner points K = imaginary quadratic field satisfying the Heegner hypothesis (HH): There exists an ideal N of O K of norm N, with O K /N Z/NZ. Definition The Heegner points on X 0 (N) of level c attached to K are the points given by pairs (A, A[N]) with End(A) = Z + co K. They are defined over the ring class field of K of conductor c. P K := π E ((A 1, A 1 [N]) + + (A h, A h [N]) h( )) E(K).

12 The Gross-Zagier Theorem Theorem (Gross-Zagier) For all K satisfying (HH), the L-series L(E/K, s) vanishes to odd order at s = 1, and L (E/K, 1) = P K, P K f, f (mod Q ). In particular, P K is of infinite order iff L (E/K, 1) 0.

13 Kolyvagin s Theorem Theorem (Kolyvagin) If P K is of infinite order, then rank(e(k)) = 1, and LI (E/K) <. The Heegner point P K 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.

14 Kolyvagin s Theorem Theorem (Kolyvagin) If P K is of infinite order, then rank(e(k)) = 1, and LI (E/K) <. The Heegner point P K 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.

15 Proof of BSD in analytic rank 1 Theorem (Gross-Zagier, Kolyvagin) If ord s=1 L(E, s) 1, then LI (E/Q) is finite and rank(e(q)) = ord s=1 L(E, s). Proof. 1. Bump-Friedberg-Hoffstein, Murty-Murty there exists a K satisfying (HH), with ord s=1 L(E/K, s) = Gross-Zagier the Heegner point P K is of infinite order. 3. Koyvagin E(K) Q = Q P K, and LI (E/K) <. 4. Explicit calculation E(Q) if L(E, 1) = 0,. the point P K belongs to E(K) if L(E, 1) 0.

16 Proof of BSD in analytic rank 1 Theorem (Gross-Zagier, Kolyvagin) If ord s=1 L(E, s) 1, then LI (E/Q) is finite and rank(e(q)) = ord s=1 L(E, s). Proof. 1. Bump-Friedberg-Hoffstein, Murty-Murty there exists a K satisfying (HH), with ord s=1 L(E/K, s) = Gross-Zagier the Heegner point P K is of infinite order. 3. Koyvagin E(K) Q = Q P K, and LI (E/K) <. 4. Explicit calculation E(Q) if L(E, 1) = 0,. the point P K belongs to E(K) if L(E, 1) 0.

17 Proof of BSD in analytic rank 1 Theorem (Gross-Zagier, Kolyvagin) If ord s=1 L(E, s) 1, then LI (E/Q) is finite and rank(e(q)) = ord s=1 L(E, s). Proof. 1. Bump-Friedberg-Hoffstein, Murty-Murty there exists a K satisfying (HH), with ord s=1 L(E/K, s) = Gross-Zagier the Heegner point P K is of infinite order. 3. Koyvagin E(K) Q = Q P K, and LI (E/K) <. 4. Explicit calculation E(Q) if L(E, 1) = 0,. the point P K belongs to E(K) if L(E, 1) 0.

18 Proof of BSD in analytic rank 1 Theorem (Gross-Zagier, Kolyvagin) If ord s=1 L(E, s) 1, then LI (E/Q) is finite and rank(e(q)) = ord s=1 L(E, s). Proof. 1. Bump-Friedberg-Hoffstein, Murty-Murty there exists a K satisfying (HH), with ord s=1 L(E/K, s) = Gross-Zagier the Heegner point P K is of infinite order. 3. Koyvagin E(K) Q = Q P K, and LI (E/K) <. 4. Explicit calculation E(Q) if L(E, 1) = 0,. the point P K belongs to E(K) if L(E, 1) 0.

19 Proof of BSD in analytic rank 1 Theorem (Gross-Zagier, Kolyvagin) If ord s=1 L(E, s) 1, then LI (E/Q) is finite and rank(e(q)) = ord s=1 L(E, s). Proof. 1. Bump-Friedberg-Hoffstein, Murty-Murty there exists a K satisfying (HH), with ord s=1 L(E/K, s) = Gross-Zagier the Heegner point P K is of infinite order. 3. Koyvagin E(K) Q = Q P K, and LI (E/K) <. 4. Explicit calculation E(Q) if L(E, 1) = 0,. the point P K belongs to E(K) if L(E, 1) 0.

20 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 of degree n > 1. Definition An elliptic curve E/F is modular if there is an automorphic representation π(e) of GL 2 (A F ) attached to E, or, equivalently, a Hilbert modular form G S 2 (N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

21 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 of degree n > 1. Definition An elliptic curve E/F is modular if there is an automorphic representation π(e) of GL 2 (A F ) attached to E, or, equivalently, a Hilbert modular form G S 2 (N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

22 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 of degree n > 1. Definition An elliptic curve E/F is modular if there is an automorphic representation π(e) of GL 2 (A F ) attached to E, or, equivalently, a Hilbert modular form G S 2 (N) over F such that L(E/F, s) = L(G, s). Modularity is often known, and will be assumed from now on.

23 Geometric modularity Geometrically, the Hilbert modular form G corresponds to a (2 n -dimensional) subspace Ω G Ω n 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 π v (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 π E : Jac(X ) E.

24 Geometric modularity Geometrically, the Hilbert modular form G corresponds to a (2 n -dimensional) subspace Ω G Ω n 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 π v (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 π E : Jac(X ) E.

25 Geometric modularity Geometrically, the Hilbert modular form G corresponds to a (2 n -dimensional) subspace Ω G Ω n 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 π v (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 π E : Jac(X ) E.

26 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 ord s=1 L(E/F, s) 1, then LI (E/F ) is finite and rank(e(f )) = ord s=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001).

27 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 ord s=1 L(E/F, s) 1, then LI (E/F ) is finite and rank(e(f )) = ord s=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001).

28 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 ord s=1 L(E/F, s) 1, then LI (E/F ) is finite and rank(e(f )) = ord s=1 L(E/F, s). Zhang, Shouwu. Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153 (2001).

29 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 LI (E/F )[p ] is finite for almost all p. Proof. Congruences between modular forms the Galois representation E[p n ] occurs in J n [p n ], where J n = Jac(X n ) and X n is a Shimura curve X n whose level may (and does) depend on n. Use CM points on X n to bound the p n -Selmer group of E.

30 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 LI (E/F )[p ] is finite for almost all p. Proof. Congruences between modular forms the Galois representation E[p n ] occurs in J n [p n ], where J n = Jac(X n ) and X n is a Shimura curve X n whose level may (and does) depend on n. Use CM points on X n to bound the p n -Selmer group of E.

31 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 LI (E/F )[p ] is finite for almost all p. Proof. Congruences between modular forms the Galois representation E[p n ] occurs in J n [p n ], where J n = Jac(X n ) and X n is a Shimura curve X n whose level may (and does) depend on n. Use CM points on X n to bound the p n -Selmer group of E.

32 The Challenge that remains When ord s=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.

33 The Challenge that remains When ord s=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.

34 The Challenge that remains When ord s=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.

35 The Challenge that remains When ord s=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.

36 ATR twists Consider the twist E M of E by a quadratic extension M/F. Proposition 1 If M is totally real or CM, then E M has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then E M has odd analytic rank. Conjecture (on ATR twists) Let M be an ATR extension of F and let E M be the associated twist of E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) <. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

37 ATR twists Consider the twist E M of E by a quadratic extension M/F. Proposition 1 If M is totally real or CM, then E M has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then E M has odd analytic rank. Conjecture (on ATR twists) Let M be an ATR extension of F and let E M be the associated twist of E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) <. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

38 ATR twists Consider the twist E M of E by a quadratic extension M/F. Proposition 1 If M is totally real or CM, then E M has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then E M has odd analytic rank. Conjecture (on ATR twists) Let M be an ATR extension of F and let E M be the associated twist of E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) <. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

39 ATR twists Consider the twist E M of E by a quadratic extension M/F. Proposition 1 If M is totally real or CM, then E M has even analytic rank. 2 If M is an ATR (Almost Totally Real) extension, then E M has odd analytic rank. Conjecture (on ATR twists) Let M be an ATR extension of F and let E M be the associated twist of E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) <. Although BSD is much better understood in analytic rank one, the conjecture on ATR twists presents a genuine mystery!

40 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 S 2 (Γ 1 (N), ε F ) 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 GL 2 (A F ) is the Doi-Naganuma lift of f. Modular parametrisation defined over F : J 1 (N) E.

41 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 S 2 (Γ 1 (N), ε F ) 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 GL 2 (A F ) is the Doi-Naganuma lift of f. Modular parametrisation defined over F : J 1 (N) E.

42 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 S 2 (Γ 1 (N), ε F ) 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 GL 2 (A F ) is the Doi-Naganuma lift of f. Modular parametrisation defined over F : J 1 (N) E.

43 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 S 2 (Γ 1 (N), ε F ) 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 GL 2 (A F ) is the Doi-Naganuma lift of f. Modular parametrisation defined over F : J 1 (N) E.

44 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 N E over a real quadratic field F, and let M/F be an ATR extension of discriminant prime to N E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) is finite.

45 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 N E over a real quadratic field F, and let M/F be an ATR extension of discriminant prime to N E. If L (E M /F, 1) 0, then E M (F ) has rank one and LI (E M /F ) is finite.

46 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.

47 Some Galois theory Let M= Galois closure of M over Q. Then Gal(M/Q) = D 8. This group contains two copies of the Klein 4-group: V F = τ M, τ M, V K = τ L, τ L. τ M τ L τ M τ L

48 Some Galois theory Suppose that F = M V F M = M τ M M = M τ M, and set K = M V K L = M τ L L = M τ L. M M M Q( N, d) L L F Q( Nd) K Q

49 Key facts about K and L Let χ M : A F ±1 be the quadratic character attached to M/F ; χ L : 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 χ L A Q 3 Ind Q F χ M = Ind Q K χ L; is equal to ε F.

50 Key facts about K and L Let χ M : A F ±1 be the quadratic character attached to M/F ; χ L : 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 χ L A Q 3 Ind Q F χ M = Ind Q K χ L; is equal to ε F.

51 Key facts about K and L Let χ M : A F ±1 be the quadratic character attached to M/F ; χ L : 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 χ L A Q 3 Ind Q F χ M = Ind Q K χ L; is equal to ε F.

52 Key facts about K and L Let χ M : A F ±1 be the quadratic character attached to M/F ; χ L : 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 χ L A Q 3 Ind Q F χ M = Ind Q K χ L; is equal to ε F.

53 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

54 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

55 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

56 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

57 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

58 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

59 The Artin formalism Let E/F be a Q-curve and let f S 2 (Γ 0 (N), ε F ) be the associated elliptic cusp form. L(E M /F, s) = L(E/F, χ M, s) = L(f /F, χ M, s) = L(f Ind Q F χ M, s) = L(f Ind Q K χ L, s) = L(f /K, χ L, s). In particular, L (E M /F, 1) 0 implies that L (f /K, χ L, 1) 0.

60 A theorem of GZK-type The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ 1 (N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L (f /K, χ L, 1) 0, then A f (L) Q has dimension one over T f, and therefore rank(a f (L) ) = 2. Furthermore LI (A f /L) is finite. rank(a f (L) ) = rank(a f (M) ), A f (M) = E(M) E(M). Corollary If L (E M /F, 1) 0, then rank(e M (F )) = 1 and LI (E M /F ) <.

61 A theorem of GZK-type The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ 1 (N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L (f /K, χ L, 1) 0, then A f (L) Q has dimension one over T f, and therefore rank(a f (L) ) = 2. Furthermore LI (A f /L) is finite. rank(a f (L) ) = rank(a f (M) ), A f (M) = E(M) E(M). Corollary If L (E M /F, 1) 0, then rank(e M (F )) = 1 and LI (E M /F ) <.

62 A theorem of GZK-type The following strikingly general recent generalisation of the GZK theorem applies to forms on Γ 1 (N) with non-trivial nebentype character. Theorem (Ye Tian, Xinyi Yuan, Shou-Wu Zhang, Wei Zhang) If L (f /K, χ L, 1) 0, then A f (L) Q has dimension one over T f, and therefore rank(a f (L) ) = 2. Furthermore LI (A f /L) is finite. rank(a f (L) ) = rank(a f (M) ), A f (M) = E(M) E(M). Corollary If L (E M /F, 1) 0, then rank(e M (F )) = 1 and LI (E M /F ) <.

63 The motivation for this study Some years ago, Adam Logan and I proposed a strategy for calculating a global point on E M (F ), based on Abel-Jacobi images of ATR cycles on Hilbert modular varieties. These so called ATR points, a particular instance of Stark-Heegner points, are constructed from periods and secondary invariants attached to Hilbert modular forms.

64 A final question Underlying my theorem with Rotger and Zhao is the construction of a classical Heegner point P M (f ) E M (F ). Question Is there a direct formula relating the classical Heegner point P M (f ) arising from J 1 (N), and the ATR point P M? (G) attached to the appropriate ATR cycle and Hilbert modular form G = DN(f )? The study undertaken with Rotger and Zhao suggests a relation of the form P? M (G)? = l P M (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.

65 A final question Underlying my theorem with Rotger and Zhao is the construction of a classical Heegner point P M (f ) E M (F ). Question Is there a direct formula relating the classical Heegner point P M (f ) arising from J 1 (N), and the ATR point P M? (G) attached to the appropriate ATR cycle and Hilbert modular form G = DN(f )? The study undertaken with Rotger and Zhao suggests a relation of the form P? M (G)? = l P M (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.

66 A final question Underlying my theorem with Rotger and Zhao is the construction of a classical Heegner point P M (f ) E M (F ). Question Is there a direct formula relating the classical Heegner point P M (f ) arising from J 1 (N), and the ATR point P M? (G) attached to the appropriate ATR cycle and Hilbert modular form G = DN(f )? The study undertaken with Rotger and Zhao suggests a relation of the form P? M (G)? = l P M (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.

67 A final question Underlying my theorem with Rotger and Zhao is the construction of a classical Heegner point P M (f ) E M (F ). Question Is there a direct formula relating the classical Heegner point P M (f ) arising from J 1 (N), and the ATR point P M? (G) attached to the appropriate ATR cycle and Hilbert modular form G = DN(f )? The study undertaken with Rotger and Zhao suggests a relation of the form P? M (G)? = l P M (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.

68 Le mot de la fin Thank you for your attention!