p -adic iterated integrals and rational points on curves Henri - PowerPoint PPT Presentation

Stark-Heegner points The completely conjectural nature of Stark-Heegner points prevents a proof of BSD ρ, 0 and BSD ρ, 1 along the lines of the proof of Kolyvagin-Goss-Zagier when ρ is induced from a character of a real quadratic field. Goal : Describe a more indirect approach whose goal is to 1 Prove BSD ρ, 0 . 2 Construct the global cohomology classes κ E ,ρ ∈ H 1 ( Q , V p ( E ) ⊗ V ρ ) which ought to arise from Stark-Heegner points via the connecting homomorphism of Kummer theory.

Stark-Heegner points The completely conjectural nature of Stark-Heegner points prevents a proof of BSD ρ, 0 and BSD ρ, 1 along the lines of the proof of Kolyvagin-Goss-Zagier when ρ is induced from a character of a real quadratic field. Goal : Describe a more indirect approach whose goal is to 1 Prove BSD ρ, 0 . 2 Construct the global cohomology classes κ E ,ρ ∈ H 1 ( Q , V p ( E ) ⊗ V ρ ) which ought to arise from Stark-Heegner points via the connecting homomorphism of Kummer theory.

p -adic deformations of geometric constructions A Λ-adic Galois representation is a finite free module V over Λ equipped with a continuous action of G Q . Specialisations : ξ ∈ W := hom cts ( Z × p , C × p ) = hom cts (Λ , C p ) , ξ : V − → V ξ := V ⊗ Λ ,ξ Q p ,ξ . Suppose there is a dense set of points Ω geom ⊂ W and, for each ξ ∈ Ω geom , a class κ ξ ∈ H 1 fin ( Q , V ξ ) . Definition The collection { κ ξ } ξ ∈ Ω geom interpolates p-adically if there exists κ ∈ H 1 ( Q , V ) such that ξ ( κ ) = κ ξ , for all ξ ∈ Ω geom .

p -adic deformations of geometric constructions A Λ-adic Galois representation is a finite free module V over Λ equipped with a continuous action of G Q . Specialisations : ξ ∈ W := hom cts ( Z × p , C × p ) = hom cts (Λ , C p ) , ξ : V − → V ξ := V ⊗ Λ ,ξ Q p ,ξ . Suppose there is a dense set of points Ω geom ⊂ W and, for each ξ ∈ Ω geom , a class κ ξ ∈ H 1 fin ( Q , V ξ ) . Definition The collection { κ ξ } ξ ∈ Ω geom interpolates p-adically if there exists κ ∈ H 1 ( Q , V ) such that ξ ( κ ) = κ ξ , for all ξ ∈ Ω geom .

p -adic deformations of geometric constructions A Λ-adic Galois representation is a finite free module V over Λ equipped with a continuous action of G Q . Specialisations : ξ ∈ W := hom cts ( Z × p , C × p ) = hom cts (Λ , C p ) , ξ : V − → V ξ := V ⊗ Λ ,ξ Q p ,ξ . Suppose there is a dense set of points Ω geom ⊂ W and, for each ξ ∈ Ω geom , a class κ ξ ∈ H 1 fin ( Q , V ξ ) . Definition The collection { κ ξ } ξ ∈ Ω geom interpolates p-adically if there exists κ ∈ H 1 ( Q , V ) such that ξ ( κ ) = κ ξ , for all ξ ∈ Ω geom .

p -adic deformations of geometric constructions A Λ-adic Galois representation is a finite free module V over Λ equipped with a continuous action of G Q . Specialisations : ξ ∈ W := hom cts ( Z × p , C × p ) = hom cts (Λ , C p ) , ξ : V − → V ξ := V ⊗ Λ ,ξ Q p ,ξ . Suppose there is a dense set of points Ω geom ⊂ W and, for each ξ ∈ Ω geom , a class κ ξ ∈ H 1 fin ( Q , V ξ ) . Definition The collection { κ ξ } ξ ∈ Ω geom interpolates p-adically if there exists κ ∈ H 1 ( Q , V ) such that ξ ( κ ) = κ ξ , for all ξ ∈ Ω geom .

p -adic limits of geometric constructions Suppose that V p ( E ) = H 1 et ( E , Q p )(1) arises as the specialisation ξ E : V − → V p ( E ) for some ξ E not necessarily belonging to Ω geom . One may then consider the class κ E := ξ E ( κ ) ∈ H 1 ( Q , V p ( E )) and attempt to relate it to L ( E , 1) and to the arithmetic of E . The class κ E is a p -adic limit of geometric classes, but need not itself admit a geometric construction.

p -adic limits of geometric constructions Suppose that V p ( E ) = H 1 et ( E , Q p )(1) arises as the specialisation ξ E : V − → V p ( E ) for some ξ E not necessarily belonging to Ω geom . One may then consider the class κ E := ξ E ( κ ) ∈ H 1 ( Q , V p ( E )) and attempt to relate it to L ( E , 1) and to the arithmetic of E . The class κ E is a p -adic limit of geometric classes, but need not itself admit a geometric construction.

p -adic limits of geometric constructions Suppose that V p ( E ) = H 1 et ( E , Q p )(1) arises as the specialisation ξ E : V − → V p ( E ) for some ξ E not necessarily belonging to Ω geom . One may then consider the class κ E := ξ E ( κ ) ∈ H 1 ( Q , V p ( E )) and attempt to relate it to L ( E , 1) and to the arithmetic of E . The class κ E is a p -adic limit of geometric classes, but need not itself admit a geometric construction.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

Basic examples Coates-Wiles : V is induced from a family of Hecke characters of a quadratic imaginary field, Ω geom = { finite order Hecke characters } , the κ ξ arise from the images of elliptic units under the Kummer map, and ξ E corresponds to a Hecke character of infinity type (1 , 0) attached to a CM elliptic curve E . Kato . V = V p ( E )(1) ⊗ Λ cyc , Ω geom = { finite order χ : Z × → C × p − p } , the κ χ ∈ H 1 ( Q , V p ( E )(1)( χ )) arise from the images of Beilinson elements in K 2 ( X 1 ( Np s )) ( s =cond( χ )), and ξ E = − 1.

The Perrin-Riou philosophy Perrin-Riou . p -adic families of global cohomology classes are a powerful tool for studying p -adic L -functions. I will illustrate this philosophy in the following contexts: 1 Classes arising from Beilinson-Kato elements, and the Mazur-Swinnerton-Dyer p -adic L -function (as described in Massimo Bertolini’s lecture); 2 Classes arising from diagonal cycles and the Harris-Tilouine triple product p -adic L -function (as discussed in Victor Rotger’s lecture).

The Perrin-Riou philosophy Perrin-Riou . p -adic families of global cohomology classes are a powerful tool for studying p -adic L -functions. I will illustrate this philosophy in the following contexts: 1 Classes arising from Beilinson-Kato elements, and the Mazur-Swinnerton-Dyer p -adic L -function (as described in Massimo Bertolini’s lecture); 2 Classes arising from diagonal cycles and the Harris-Tilouine triple product p -adic L -function (as discussed in Victor Rotger’s lecture).

The Perrin-Riou philosophy Perrin-Riou . p -adic families of global cohomology classes are a powerful tool for studying p -adic L -functions. I will illustrate this philosophy in the following contexts: 1 Classes arising from Beilinson-Kato elements, and the Mazur-Swinnerton-Dyer p -adic L -function (as described in Massimo Bertolini’s lecture); 2 Classes arising from diagonal cycles and the Harris-Tilouine triple product p -adic L -function (as discussed in Victor Rotger’s lecture).

The Perrin-Riou philosophy Perrin-Riou . p -adic families of global cohomology classes are a powerful tool for studying p -adic L -functions. I will illustrate this philosophy in the following contexts: 1 Classes arising from Beilinson-Kato elements, and the Mazur-Swinnerton-Dyer p -adic L -function (as described in Massimo Bertolini’s lecture); 2 Classes arising from diagonal cycles and the Harris-Tilouine triple product p -adic L -function (as discussed in Victor Rotger’s lecture).

Modular units Y 1 ( N ) / C / C × has “maximal possible Manin-Drinfeld : the group O × rank”, namely #( X 1 ( N ) − Y 1 ( N )) − 1. The logarithmic derivative gives a surjective map dlog : O × Y 1 ( N ) / Q ( µ N ) ⊗ Q − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series. Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = G 2 ,χ , ∞ G 2 ,χ = 2 − 1 L ( χ, − 1) + � � σ χ ( n ) q n , σ χ ( n ) = χ ( d ) d . n =1 d | n

Modular units Y 1 ( N ) / C / C × has “maximal possible Manin-Drinfeld : the group O × rank”, namely #( X 1 ( N ) − Y 1 ( N )) − 1. The logarithmic derivative gives a surjective map dlog : O × Y 1 ( N ) / Q ( µ N ) ⊗ Q − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series. Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = G 2 ,χ , ∞ G 2 ,χ = 2 − 1 L ( χ, − 1) + � � σ χ ( n ) q n , σ χ ( n ) = χ ( d ) d . n =1 d | n

Modular units Y 1 ( N ) / C / C × has “maximal possible Manin-Drinfeld : the group O × rank”, namely #( X 1 ( N ) − Y 1 ( N )) − 1. The logarithmic derivative gives a surjective map dlog : O × Y 1 ( N ) / Q ( µ N ) ⊗ Q − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series. Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = G 2 ,χ , ∞ G 2 ,χ = 2 − 1 L ( χ, − 1) + � � σ χ ( n ) q n , σ χ ( n ) = χ ( d ) d . n =1 d | n

Modular units Y 1 ( N ) / C / C × has “maximal possible Manin-Drinfeld : the group O × rank”, namely #( X 1 ( N ) − Y 1 ( N )) − 1. The logarithmic derivative gives a surjective map dlog : O × Y 1 ( N ) / Q ( µ N ) ⊗ Q − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series. Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = G 2 ,χ , ∞ G 2 ,χ = 2 − 1 L ( χ, − 1) + � � σ χ ( n ) q n , σ χ ( n ) = χ ( d ) d . n =1 d | n

Modular units Y 1 ( N ) / C / C × has “maximal possible Manin-Drinfeld : the group O × rank”, namely #( X 1 ( N ) − Y 1 ( N )) − 1. The logarithmic derivative gives a surjective map dlog : O × Y 1 ( N ) / Q ( µ N ) ⊗ Q − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series. Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = G 2 ,χ , ∞ G 2 ,χ = 2 − 1 L ( χ, − 1) + � � σ χ ( n ) q n , σ χ ( n ) = χ ( d ) d . n =1 d | n

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Beilinson elements Given χ of conductor Np s , H 1 et ( X 1 ( Np s ) , Z p (1)) , α χ := δ ( u χ ) ∈ H 1 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (1)) β χ := δ ( w ζ u χ ) ∈ H 2 et ( X 1 ( Np s ) Q ( µ Nps ) , Z p (2)) , κ χ := α χ ∪ β χ ˜ ∈ H 1 ( Q ( µ Np s ) , H 1 et ( X 1 ( Np s ) ¯ κ χ := its image in Q , Z p (2))) . The latter descends to a class κ χ ∈ H 1 ( Q , H 1 et ( X 1 ( Np s ) ¯ Q , Z p (2))( χ − 1 )) . Let X 1 ( N ) − → E be a modular elliptic curve, and κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) be the natural image.

Kato’s Λ-adic class Key Remark : The Eisenstein series G 2 ,χ 0 χ (with f χ = p s ) are the weight two specialisations of a Hida family G χ 0 . Theorem (Kato) There is a Λ -adic cohomology class κ E ( G χ 0 , G χ 0 ) ∈ H 1 ( Q , V p ( E )( χ 0 ) ⊗ Λ cyc ( − 1)) , satisfying ξ 2 ,χ ( κ E ( G χ 0 , G χ 0 )) = α p ( E ) − s κ E ( G 2 ,χ 0 χ , G 2 ,χ 0 χ ) at all ”weight two” specialisations ξ 2 ,χ .

Kato’s Λ-adic class Key Remark : The Eisenstein series G 2 ,χ 0 χ (with f χ = p s ) are the weight two specialisations of a Hida family G χ 0 . Theorem (Kato) There is a Λ -adic cohomology class κ E ( G χ 0 , G χ 0 ) ∈ H 1 ( Q , V p ( E )( χ 0 ) ⊗ Λ cyc ( − 1)) , satisfying ξ 2 ,χ ( κ E ( G χ 0 , G χ 0 )) = α p ( E ) − s κ E ( G 2 ,χ 0 χ , G 2 ,χ 0 χ ) at all ”weight two” specialisations ξ 2 ,χ .

The Kato–Perrin-Riou class We can now specialise the Λ-adic cohomology class κ E ( G χ 0 , G χ 0 ) to Eisenstein series of weight one . κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) := ν 1 ( κ E ( G χ 0 , G χ 0 )) . Theorem (Kato) The class κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) is cristalline if and only if L ( E , 1) L ( E , χ − 1 0 , 1) = 0 . Corollary BSD χ, 0 is true for E.

The Kato–Perrin-Riou class We can now specialise the Λ-adic cohomology class κ E ( G χ 0 , G χ 0 ) to Eisenstein series of weight one . κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) := ν 1 ( κ E ( G χ 0 , G χ 0 )) . Theorem (Kato) The class κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) is cristalline if and only if L ( E , 1) L ( E , χ − 1 0 , 1) = 0 . Corollary BSD χ, 0 is true for E.

The Kato–Perrin-Riou class We can now specialise the Λ-adic cohomology class κ E ( G χ 0 , G χ 0 ) to Eisenstein series of weight one . κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) := ν 1 ( κ E ( G χ 0 , G χ 0 )) . Theorem (Kato) The class κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) is cristalline if and only if L ( E , 1) L ( E , χ − 1 0 , 1) = 0 . Corollary BSD χ, 0 is true for E.

The Kato–Perrin-Riou class We can now specialise the Λ-adic cohomology class κ E ( G χ 0 , G χ 0 ) to Eisenstein series of weight one . κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) := ν 1 ( κ E ( G χ 0 , G χ 0 )) . Theorem (Kato) The class κ E ( G 1 ,χ 0 , G 1 ,χ 0 ) is cristalline if and only if L ( E , 1) L ( E , χ − 1 0 , 1) = 0 . Corollary BSD χ, 0 is true for E.

Hida families To prove BSD ρ, 0 for larger classes of ρ , we will 1 replace the Beilinson elements κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) by geometric elements κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ( k − 1)) attached to a pair of cusp forms g and h of the same weight k ≥ 2. 2 Interpolate these classes in Hida families → κ E ( g , h ). 3 Consider the weight one specialisations κ E ( g 1 , h 1 ) ∈ H 1 ( Q , V p ( E ) ⊗ V ρ g 1 ⊗ V ρ h 1 ) . Of special interest is the case where ρ g 1 and ρ g 2 are Artin representations .

Hida families To prove BSD ρ, 0 for larger classes of ρ , we will 1 replace the Beilinson elements κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) by geometric elements κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ( k − 1)) attached to a pair of cusp forms g and h of the same weight k ≥ 2. 2 Interpolate these classes in Hida families → κ E ( g , h ). 3 Consider the weight one specialisations κ E ( g 1 , h 1 ) ∈ H 1 ( Q , V p ( E ) ⊗ V ρ g 1 ⊗ V ρ h 1 ) . Of special interest is the case where ρ g 1 and ρ g 2 are Artin representations .

Hida families To prove BSD ρ, 0 for larger classes of ρ , we will 1 replace the Beilinson elements κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) by geometric elements κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ( k − 1)) attached to a pair of cusp forms g and h of the same weight k ≥ 2. 2 Interpolate these classes in Hida families → κ E ( g , h ). 3 Consider the weight one specialisations κ E ( g 1 , h 1 ) ∈ H 1 ( Q , V p ( E ) ⊗ V ρ g 1 ⊗ V ρ h 1 ) . Of special interest is the case where ρ g 1 and ρ g 2 are Artin representations .

Hida families To prove BSD ρ, 0 for larger classes of ρ , we will 1 replace the Beilinson elements κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) by geometric elements κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ( k − 1)) attached to a pair of cusp forms g and h of the same weight k ≥ 2. 2 Interpolate these classes in Hida families → κ E ( g , h ). 3 Consider the weight one specialisations κ E ( g 1 , h 1 ) ∈ H 1 ( Q , V p ( E ) ⊗ V ρ g 1 ⊗ V ρ h 1 ) . Of special interest is the case where ρ g 1 and ρ g 2 are Artin representations .

Hida families To prove BSD ρ, 0 for larger classes of ρ , we will 1 replace the Beilinson elements κ E ( G 2 ,χ , G 2 ,χ ) ∈ H 1 ( Q , V p ( E )(1)( χ − 1 )) by geometric elements κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ( k − 1)) attached to a pair of cusp forms g and h of the same weight k ≥ 2. 2 Interpolate these classes in Hida families → κ E ( g , h ). 3 Consider the weight one specialisations κ E ( g 1 , h 1 ) ∈ H 1 ( Q , V p ( E ) ⊗ V ρ g 1 ⊗ V ρ h 1 ) . Of special interest is the case where ρ g 1 and ρ g 2 are Artin representations .

Gross-Kudla-Schoen diagonal classes etale Abel-Jacobi map : ´ AJ et : CH 2 ( X 1 ( N ) 3 ) 0 H 4 et ( X 1 ( N ) 3 , Q p (2)) 0 − → 3 , Q p (2))) H 1 ( Q , H 3 − → et ( X 1 ( N ) H 1 ( Q , H 1 et ( X 1 ( N ) , Q p ) ⊗ 3 (2)) − → Gross-Kudla Schoen class: κ E ( g , h ) := AJ et (∆) f , g , h ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h (1)) .

Gross-Kudla-Schoen diagonal classes etale Abel-Jacobi map : ´ AJ et : CH 2 ( X 1 ( N ) 3 ) 0 H 4 et ( X 1 ( N ) 3 , Q p (2)) 0 − → 3 , Q p (2))) H 1 ( Q , H 3 − → et ( X 1 ( N ) H 1 ( Q , H 1 et ( X 1 ( N ) , Q p ) ⊗ 3 (2)) − → Gross-Kudla Schoen class: κ E ( g , h ) := AJ et (∆) f , g , h ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h (1)) .

Gross-Kudla-Schoen diagonal classes etale Abel-Jacobi map : ´ AJ et : CH 2 ( X 1 ( N ) 3 ) 0 H 4 et ( X 1 ( N ) 3 , Q p (2)) 0 − → 3 , Q p (2))) H 1 ( Q , H 3 − → et ( X 1 ( N ) H 1 ( Q , H 1 et ( X 1 ( N ) , Q p ) ⊗ 3 (2)) − → Gross-Kudla Schoen class: κ E ( g , h ) := AJ et (∆) f , g , h ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h (1)) .

Gross-Kudla-Schoen diagonal classes etale Abel-Jacobi map : ´ AJ et : CH 2 ( X 1 ( N ) 3 ) 0 H 4 et ( X 1 ( N ) 3 , Q p (2)) 0 − → 3 , Q p (2))) H 1 ( Q , H 3 − → et ( X 1 ( N ) H 1 ( Q , H 1 et ( X 1 ( N ) , Q p ) ⊗ 3 (2)) − → Gross-Kudla Schoen class: κ E ( g , h ) := AJ et (∆) f , g , h ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h (1)) .

Hida Families Weight space : Ω := hom(Λ , C p ) ⊂ hom((1 + p Z p ) × , C × p ) . The integers form a dense subset of Ω via k ↔ ( x �→ x k ). Classical weights : Ω cl := Z ≥ 2 ⊂ Ω . If ˜ Λ is a finite flat extension of Λ, let ˜ X = hom(˜ Λ , C p ) and let κ : ˜ X − → Ω be the natural projection to weight space. Classical points : ˜ X cl := { x ∈ ˜ X such that κ ( x ) ∈ Ω cl } .

Hida Families Weight space : Ω := hom(Λ , C p ) ⊂ hom((1 + p Z p ) × , C × p ) . The integers form a dense subset of Ω via k ↔ ( x �→ x k ). Classical weights : Ω cl := Z ≥ 2 ⊂ Ω . If ˜ Λ is a finite flat extension of Λ, let ˜ X = hom(˜ Λ , C p ) and let κ : ˜ X − → Ω be the natural projection to weight space. Classical points : ˜ X cl := { x ∈ ˜ X such that κ ( x ) ∈ Ω cl } .

Hida Families Weight space : Ω := hom(Λ , C p ) ⊂ hom((1 + p Z p ) × , C × p ) . The integers form a dense subset of Ω via k ↔ ( x �→ x k ). Classical weights : Ω cl := Z ≥ 2 ⊂ Ω . If ˜ Λ is a finite flat extension of Λ, let ˜ X = hom(˜ Λ , C p ) and let κ : ˜ X − → Ω be the natural projection to weight space. Classical points : ˜ X cl := { x ∈ ˜ X such that κ ( x ) ∈ Ω cl } .

Hida Families Weight space : Ω := hom(Λ , C p ) ⊂ hom((1 + p Z p ) × , C × p ) . The integers form a dense subset of Ω via k ↔ ( x �→ x k ). Classical weights : Ω cl := Z ≥ 2 ⊂ Ω . If ˜ Λ is a finite flat extension of Λ, let ˜ X = hom(˜ Λ , C p ) and let κ : ˜ X − → Ω be the natural projection to weight space. Classical points : ˜ X cl := { x ∈ ˜ X such that κ ( x ) ∈ Ω cl } .

Hida Families Weight space : Ω := hom(Λ , C p ) ⊂ hom((1 + p Z p ) × , C × p ) . The integers form a dense subset of Ω via k ↔ ( x �→ x k ). Classical weights : Ω cl := Z ≥ 2 ⊂ Ω . If ˜ Λ is a finite flat extension of Λ, let ˜ X = hom(˜ Λ , C p ) and let κ : ˜ X − → Ω be the natural projection to weight space. Classical points : ˜ X cl := { x ∈ ˜ X such that κ ( x ) ∈ Ω cl } .

Hida families, cont’d Definition A Hida family of tame level N is a triple (Λ , Ω , g ) , where 1 Λ g is a finite flat extension of Λ; 2 Ω g ⊂ X g := hom(Λ g , C p ) is a non-empty open subset (for the p -adic topology); n a n q n ∈ Λ g [[ q ]] is a formal q -series, such that 3 g = � n x ( a n ) q n is the q series of the ordinary g ( x ) := � p-stabilisation g ( p ) of a normalised eigenform, denoted g x , of x weight κ ( x ) on Γ 1 ( N ), for all x ∈ Ω g , cl := Ω g ∩ X g , cl .

Hida families, cont’d Definition A Hida family of tame level N is a triple (Λ , Ω , g ) , where 1 Λ g is a finite flat extension of Λ; 2 Ω g ⊂ X g := hom(Λ g , C p ) is a non-empty open subset (for the p -adic topology); n a n q n ∈ Λ g [[ q ]] is a formal q -series, such that 3 g = � n x ( a n ) q n is the q series of the ordinary g ( x ) := � p-stabilisation g ( p ) of a normalised eigenform, denoted g x , of x weight κ ( x ) on Γ 1 ( N ), for all x ∈ Ω g , cl := Ω g ∩ X g , cl .

Hida families, cont’d Definition A Hida family of tame level N is a triple (Λ , Ω , g ) , where 1 Λ g is a finite flat extension of Λ; 2 Ω g ⊂ X g := hom(Λ g , C p ) is a non-empty open subset (for the p -adic topology); n a n q n ∈ Λ g [[ q ]] is a formal q -series, such that 3 g = � n x ( a n ) q n is the q series of the ordinary g ( x ) := � p-stabilisation g ( p ) of a normalised eigenform, denoted g x , of x weight κ ( x ) on Γ 1 ( N ), for all x ∈ Ω g , cl := Ω g ∩ X g , cl .

Hida families, cont’d Definition A Hida family of tame level N is a triple (Λ , Ω , g ) , where 1 Λ g is a finite flat extension of Λ; 2 Ω g ⊂ X g := hom(Λ g , C p ) is a non-empty open subset (for the p -adic topology); n a n q n ∈ Λ g [[ q ]] is a formal q -series, such that 3 g = � n x ( a n ) q n is the q series of the ordinary g ( x ) := � p-stabilisation g ( p ) of a normalised eigenform, denoted g x , of x weight κ ( x ) on Γ 1 ( N ), for all x ∈ Ω g , cl := Ω g ∩ X g , cl .

Λ-adic Galois representations If g and h are Hida families, there are associated Λ-adic Galois representations V g and V h of rank two over Λ g and Λ h respectively (cf. Adrian Iovita’s lecture on Thursday).

A p -adic family of global classes Theorem (Rotger-D) Let g and h be two Hida families. There is a Λ g ⊗ Λ Λ h -adic cohomology class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ( V g ⊗ Λ V h ) ⊗ Λ Λ cyc ( − 1)) , where V g , V h = Hida’s Λ -adic representations attached to g and h, satisfying, for all ”weight two” points ( y , z ) ∈ Ω g × Ω h , ξ y , z ( κ E ( g , h )) = ∗ κ E ( g y , h z ) . This Λ-adic class generalises Kato’s class, which one recovers when g and h are Hida families of Eisenstein series.

A p -adic family of global classes Theorem (Rotger-D) Let g and h be two Hida families. There is a Λ g ⊗ Λ Λ h -adic cohomology class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ( V g ⊗ Λ V h ) ⊗ Λ Λ cyc ( − 1)) , where V g , V h = Hida’s Λ -adic representations attached to g and h, satisfying, for all ”weight two” points ( y , z ) ∈ Ω g × Ω h , ξ y , z ( κ E ( g , h )) = ∗ κ E ( g y , h z ) . This Λ-adic class generalises Kato’s class, which one recovers when g and h are Hida families of Eisenstein series.

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

Generalised Kato Classes Lei, Loeffler and Zerbes are studying similar families of “twisted Beilinson-Flach elements”. There is a strong parallel between the three settings: 1 Beilinson-Kato elements , leading to the Kato class κ E ( G 1 ,χ , G 1 ,χ ) ∈ H 1 ( Q , V p ( E )( χ − 1 )); 2 Twisted diagonal cycles , leading to classes κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ⊗ V h ) where g and h are cusp forms of weight one with det( V g ⊗ V h ) = 1; 3 The twisted Beilinson-Flach elements in David Loeffler’s lecture, leading to classes κ E ( g , G 1 ,χ ) ∈ H 1 ( Q , V p ( E ) ⊗ V g ), where V g is a not-necessarily-self-dual representation. All three will be called generalised Kato classes for E .

A reciprocity law for diagonal cycles As in Kato’s reciprocity law, one can consider the specialisations of κ E ( g , h ) when g and h are evaluated at points of weight one . Theorem (Rotger-D; still in progress) Let ( y , z ) ∈ Ω g × Ω h be points with wt ( y ) = wt ( z ) = 1 . The class κ E ( g y , h z ) is cristalline if and only if L ( V p ( E ) ⊗ g y ⊗ h z , 1) = 0 . Main ingredients : 1. The p -adic Gross-Zagier formula for diagonal cycles described in Rotger’s lecture, and its extension to levels divisible by powers of p ; 2. Perrin-Riou’s theory of Bloch-Kato logarithms and dual exponential maps “in p -adic families”.

A reciprocity law for diagonal cycles As in Kato’s reciprocity law, one can consider the specialisations of κ E ( g , h ) when g and h are evaluated at points of weight one . Theorem (Rotger-D; still in progress) Let ( y , z ) ∈ Ω g × Ω h be points with wt ( y ) = wt ( z ) = 1 . The class κ E ( g y , h z ) is cristalline if and only if L ( V p ( E ) ⊗ g y ⊗ h z , 1) = 0 . Main ingredients : 1. The p -adic Gross-Zagier formula for diagonal cycles described in Rotger’s lecture, and its extension to levels divisible by powers of p ; 2. Perrin-Riou’s theory of Bloch-Kato logarithms and dual exponential maps “in p -adic families”.

A reciprocity law for diagonal cycles As in Kato’s reciprocity law, one can consider the specialisations of κ E ( g , h ) when g and h are evaluated at points of weight one . Theorem (Rotger-D; still in progress) Let ( y , z ) ∈ Ω g × Ω h be points with wt ( y ) = wt ( z ) = 1 . The class κ E ( g y , h z ) is cristalline if and only if L ( V p ( E ) ⊗ g y ⊗ h z , 1) = 0 . Main ingredients : 1. The p -adic Gross-Zagier formula for diagonal cycles described in Rotger’s lecture, and its extension to levels divisible by powers of p ; 2. Perrin-Riou’s theory of Bloch-Kato logarithms and dual exponential maps “in p -adic families”.

A reciprocity law for diagonal cycles As in Kato’s reciprocity law, one can consider the specialisations of κ E ( g , h ) when g and h are evaluated at points of weight one . Theorem (Rotger-D; still in progress) Let ( y , z ) ∈ Ω g × Ω h be points with wt ( y ) = wt ( z ) = 1 . The class κ E ( g y , h z ) is cristalline if and only if L ( V p ( E ) ⊗ g y ⊗ h z , 1) = 0 . Main ingredients : 1. The p -adic Gross-Zagier formula for diagonal cycles described in Rotger’s lecture, and its extension to levels divisible by powers of p ; 2. Perrin-Riou’s theory of Bloch-Kato logarithms and dual exponential maps “in p -adic families”.

BSD ρ in analytic rank zero. Corollary Let E be an elliptic curve over Q and ρ 1 , ρ 2 odd irreducible two-dimensional Galois representations. Then BSD ρ 1 ⊗ ρ 2 , 0 is true for E. Proof . Use the ramified class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ρ 1 ⊗ ρ 2 ) to bound the image of the global points in the local points. Corollary Let χ be a dihedral character of a real quadratic field K, and let ρ = Ind Q K χ . Then BSD ρ, 0 is true. Proof . Specialise to the case ρ 1 = Ind Q F χ 1 and ρ 2 = Ind Q F χ 2 .

BSD ρ in analytic rank zero. Corollary Let E be an elliptic curve over Q and ρ 1 , ρ 2 odd irreducible two-dimensional Galois representations. Then BSD ρ 1 ⊗ ρ 2 , 0 is true for E. Proof . Use the ramified class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ρ 1 ⊗ ρ 2 ) to bound the image of the global points in the local points. Corollary Let χ be a dihedral character of a real quadratic field K, and let ρ = Ind Q K χ . Then BSD ρ, 0 is true. Proof . Specialise to the case ρ 1 = Ind Q F χ 1 and ρ 2 = Ind Q F χ 2 .

BSD ρ in analytic rank zero. Corollary Let E be an elliptic curve over Q and ρ 1 , ρ 2 odd irreducible two-dimensional Galois representations. Then BSD ρ 1 ⊗ ρ 2 , 0 is true for E. Proof . Use the ramified class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ρ 1 ⊗ ρ 2 ) to bound the image of the global points in the local points. Corollary Let χ be a dihedral character of a real quadratic field K, and let ρ = Ind Q K χ . Then BSD ρ, 0 is true. Proof . Specialise to the case ρ 1 = Ind Q F χ 1 and ρ 2 = Ind Q F χ 2 .

BSD ρ in analytic rank zero. Corollary Let E be an elliptic curve over Q and ρ 1 , ρ 2 odd irreducible two-dimensional Galois representations. Then BSD ρ 1 ⊗ ρ 2 , 0 is true for E. Proof . Use the ramified class κ E ( g , h ) ∈ H 1 ( Q , V p ( E ) ⊗ ρ 1 ⊗ ρ 2 ) to bound the image of the global points in the local points. Corollary Let χ be a dihedral character of a real quadratic field K, and let ρ = Ind Q K χ . Then BSD ρ, 0 is true. Proof . Specialise to the case ρ 1 = Ind Q F χ 1 and ρ 2 = Ind Q F χ 2 .

Analytic rank one, and Stark-Heegner points? Question . Assume that 1 g and h are attached to classical modular forms, and hence to Artin representations ρ g and ρ h ; 2 L ( E , ρ g ⊗ ρ h , 1) = 0, so that κ E ( g , h ) is cristalline. Project with Lauder and Rotger : Give an explicit, computable formula for log p ( κ E ( g , h )) ∈ (Ω 1 ( E / Q p ) ⊗ D ( V ρ g ) ⊗ D ( V ρ h )) ∨ . This would be useful both for theoretical and experimental purposes.