Diagonal cycles and Euler systems for real quadratic fields Henri - PowerPoint PPT Presentation

Diagonal cycles and Euler systems for real quadratic fields Henri Darmon An ongoing project with Victor Rotger Conference on the Birch and Swinnerton-Dyer Conjecture, Cambridge, UK May 2011

The Birch and Swinnerton-Dyer conjecture One of the major outstanding issues in the Birch and Swinnerton-Dyer conjecture is the (explicit) construction of rational points on elliptic curves. There are very few strategies for producing such rational points: 1 Heegner points (CM points on modular elliptic curves). Birch, Gross-Zagier-Zhang, Kolyvagin... 2 Higher-dimensional algebraic cycles can sometimes be used to construct “interesting” rational points on elliptic curves, as described in Victor’s lecture.

Summary of Victor’s Lecture Cycle classes in the triple product of modular curves lead to rational points on elliptic curves. These points make it possible to relate: 1 Certain extension classes (of mixed motives) arising in the pro-unipotent fundamental groups of modular curves; 2 Special values of L -functions of modular forms. This fits into a general philosophy (Deligne, Wojtkowiak, ...) relating π unip ( X ) to values of L -functions. 1

What about BSD? Question : Do these constructions yield anything “genuinely new” about the Birch and Swinnerton-Dyer conjecture for elliptic curves over Q ? BSD Conjecture : r an ( E / Q ) = r ( E / Q ), where r an ( E / Q ) := ord s =1 L ( E / Q , s ) , r ( E / Q ) = rank( E ( Q )) . r an ( E / Q ) ≤ 1: everything is known. r an ( E / Q ) > 1: we haven’t the slightest idea.

A “equivariant” BSD conjecture L -functions carry a lot of information about the structure of E (¯ Q ). Consider a continuous Artin representation ρ : Gal ( K ρ / Q ) − → GL n ( C ) . r an ( E , ρ ) := ord s =1 L ( E , ρ, s ) , r ( E , ρ ) := dim C hom( V ρ , E ( K ρ ) ⊗ C ) . Conjecture (“Equivariant” BSD) For all Artin representations ρ , r an ( E , ρ ) = r ( E , ρ ) .

What is known? The following cases of the conjecture have been established: 1 ρ is one-dimensional (i.e., corresponds to a Dirichlet character), and r an ( E , ρ ) = 0. (Kato, 1991). 2 ρ = Ind Q K χ , where χ =dihedral, K =quadratic imaginary field, r an ( E , ρ ) = 1. (Kolyvagin, Gross-Zagier, Zhang, ...., 1989). 3 Similar setting, r an ( E , ρ ) = 0. (Bertolini-D, Rotger, Vigni, Nekovar,... ,1996).

Artin Representations We will be primarily interested in odd Artin representations ρ : Gal ( K ρ / Q ) − → GL 2 ( C ) . The cases that can arise are: 1 ρ = Ind Q K χ , where K = imaginary quadratic field; 2 ρ = Ind Q F χ , where F = real quadratic field, and χ has signature (+ , − ). 3 The projective image of ρ is A 4 , S 4 or A 5 .

The BSD theorem E = elliptic curve over Q ; ρ 1 , ρ 2 = odd 2-dimensional representations of G Q , det( ρ 1 ) det( ρ 2 ) = 1 . The following theorem is the the primary goal of the current project with V. Rotger. Theorem (Rotger, D: still in progress, and far from complete!) Assume that there exists σ ∈ G Q for which ρ 1 ⊗ ρ 2 ( σ ) has distinct eigenvalues. If L ( E ⊗ ρ 1 ⊗ ρ 2 , 1) � = 0 , then hom( V ρ 1 ⊗ V ρ 2 , E ( K ρ 1 K ρ 2 ) ⊗ C ) = 0 .

Modularity The objects E , ρ 1 , and ρ 2 are all known to be modular! As usual, this plays a key role. Theorem (Hecke, Langlands-Tunnell, Wiles, Taylor, Khare,. . . ) There exist modular forms f of weight two, and g and h of weight one, such that L ( f , s ) = L ( E , s ) , L ( g , s ) = L ( ρ 1 , s ) , L ( h , s ) = L ( ρ 2 , s ) .

Strategy of the proof The strategy for the proof of our sought-for Theorem rests on the following key ingredients. 1 Galois cohomology classes κ ( f , g ′ , h ′ ) ∈ H 1 ( Q , V f ⊗ V g ′ ⊗ V h ′ ) attached to a triple ( f , g ′ , h ′ ) of modular forms of weights ≥ 2. They are obtained from the image of diagonal cycles on triple products of Kuga-Sato varieties under p -adic ´ etale Abel-Jacobi maps. 2 p -adic deformations of these classes, attached to Hida families f , g and h interpolating f , g and h . 3 Various relations between these classes and L -functions (both complex and p -adic) attached to f ⊗ g ⊗ h .

Triples of eigenforms Definition A triple of eigenforms f ∈ S k (Γ 0 ( N f ) , ε f ) , g ∈ S ℓ (Γ 0 ( N g ) , ε g ) , h ∈ S m (Γ 0 ( N h ) , ε h ) is said to be critical if 1 Their weights are balanced : k < ℓ + m , ℓ < k + m , m < k + ℓ. 2 ε f ε g ε h = 1; in particular, k + ℓ + m is even.

Diagonal cycles on triple products of Kuga-Sato varieties. r = r 1 + r 2 + r 3 k = r 1 + 2 , ℓ = r 2 + 2 , m = r 3 + 2 , . 2 E r ( N ) = r -fold Kuga-Sato variety over X 1 ( N ); dim = r + 1 . V = E r 1 ( N f ) × E r 2 ( N g ) × E r 3 ( N h ) , dim V = 2 r + 3 . Generalised Gross-Kudla-Schoen cycle : there is an essentially unique interesting way of embedding E r ( N f N g N h ) as a null-homologous cycle in V . Cf. Rotger, D. Notes for the AWS, Chapter 7. ∆ = E r ⊂ V , ∆ ∈ C H r +2 ( V ) .

Diagonal cycles and L -series The height of the ( f , g , h )-isotypic component ∆ f , g , h of the Gross-Kudla-Schoen cycle ∆ should be related to the central critical derivative L ′ ( f ⊗ g ⊗ h , r + 2) . Work of Yuan-Zhang-Zhang represents substantial progress in this direction, when r 1 = r 2 = r 3 = 0. Our goal will be instead: to describe other relationships between ∆ f , g , h and p -adic L -series attached to ( f , g , h ), in view of obtaining the arithmetic applications described above.

Complex Abel-Jacobi maps The cycle ∆ is null-homologous: cl (∆) = 0 in H 2 r +4 ( V ( C ) , Q ) . Our formula of “Gross-Kudla-Zhang type” will not involve heights, but rather p -adic analogues of the complex Abel-Jacobi map of Griffiths and Weil: H 2 r +3 ( V / C ) AJ : CH r +2 ( V ) 0 dR − → Fil r +2 H 2 r +3 ( V / C ) + H 2 r +3 ( V ( C ) , Z ) dR B = Fil r +2 H 2 r +3 ( V / C ) ∨ dR . H 2 r +3 ( V ( C ) , Z ) � AJ(∆)( ω ) = ω. ∂ − 1 ∆

� � � � p -adic ´ etale Abel-Jacobi maps AJ et CH r +2 ( V / Q ) 0 f ( Q , H 2 r +3 ( ¯ H 1 V , Q p )( r + 2)) et � H 1 CH r +2 ( V / Q p ) 0 f ( Q p , H 2 r +3 ( ¯ V , Q p )( r + 2)) et AJ et Fil r +2 H 2 r +3 ( V / Q p ) ∨ dR The dotted arrow is called the p-adic Abel-Jacobi map and denoted AJ p . Goal : Relate AJ p (∆) to certain Rankin triple product p -adic L -functions, ` a la Gross-Kudla-Zhang.

Hida families Let p be any prime, and replace f , g and h by their p -stabilisations, which are both ordinary (eigenvectors for U p with eigenvalue a p -adic unit). Theorem (Hida) There exist p-adic families � � � a n ( k ) q n , b n ( ℓ ) q n , c n ( m ) q n , f ( k ) = g ( ℓ ) = h ( m ) = (k , ℓ, m ∈ Z / ( p − 1) Z × Z p ) of modular forms satisfying f (2) = f , g (1) = g and h (1) = h. For k , ℓ, m ∈ Z ≥ 2 , the specialisations f k := f ( k ) , g ℓ := g ( ℓ ) , h m := h ( m ) are classical eigenforms of weights k , ℓ and m .

Triple product p -adic Rankin L -functions They interpolate the central critical values L ( f ( k ) ⊗ g ( ℓ ) ⊗ h ( m ) , k + ℓ + m − 2 ) 2 ∈ ¯ Q . Ω( k , ℓ, m ) Four distinct regions of interpolation: 1 Σ f = { ( k , ℓ, m ) : k ≥ ℓ + m } . Ω( k , ℓ, m ) = ∗� f k , f k � 2 . 2 Σ g = { ( k , ℓ, m ) : ℓ ≥ k + m } . Ω( k , ℓ, m ) = ∗� g ℓ , g ℓ � 2 . 3 Σ h = { ( k , ℓ, m ) : m ≥ k + ℓ } . Ω( k , ℓ, m ) = ∗� h m , h m � 2 . 4 Σ bal = ( Z ≥ 2 ) 3 − Σ f − Σ g − Σ h . Ω( k , ℓ, m ) = ∗� f k , f k � 2 � g ℓ , g ℓ � 2 � h m , h m � 2 . Resulting p -adic L -functions: L f p ( f ⊗ g ⊗ h , k , ℓ, m ), L g p ( f ⊗ g ⊗ h , k , ℓ, m ), and L h p ( f ⊗ g ⊗ h , k , ℓ, m ) respectively.

More notations ω f = (2 π i ) r 1 +1 f ( τ ) dw 1 · · · dw r 1 d τ ∈ Fil r 1 +1 H r 1 +1 ( E r 1 ). dR η f ∈ H r 1 +1 ( E r 1 / ¯ Q p ) = representative of the f -isotypic part on dR which Frobenius acts via α p ( f ), normalised so that � ω f , η f � = 1 . Lemma If ( k , ℓ, m ) is balanced, then the ( f k , g ℓ , h m ) -isotypic part of the ¯ Q p vector space Fil r +2 H 2 r +3 ( V / ¯ Q p ) is generated by the classes of dR ω f k ⊗ ω g ℓ ⊗ ω h m , η f k ⊗ ω g ℓ ⊗ ω h m , ω f k ⊗ η g ℓ ⊗ ω h m , ω f k ⊗ ω g ℓ ⊗ η h m .

A p -adic Gross-Kudla formula Assume that sign ( L ( f k ⊗ g ℓ ⊗ h m , s )) = − 1 for all ( k , ℓ, m ) ∈ Σ bal . (For example, f , g and h are of the same level.) Theorem (Rotger-Sols-D; in progress) For all ( k , ℓ, m ) ∈ Σ bal , L f p ( f ⊗ g ⊗ h , k , ℓ, m ) = ∗ × AJ p (∆)( η f ∧ ω g ∧ ω h ) , and likewise for L g p and L h p . Conclusion : The Abel-Jacobi image of ∆ encodes the special values of the three distinct p -adic L -functions.

From cycles to cohomology classes We can use the cycles ∆ k ,ℓ, m to construct global classes AJ et (∆ k ,ℓ, m ) ∈ H 1 ( Q , H 2 r +3 ( V ¯ Q , Q p )( r + 2)) . et K¨ unneth: 3 H r j +1 ( E r j � H 2 r +3 ( V ¯ Q , Q p )( r + 2) → Q , Q p )( r + 2) et et ¯ j =1 → V f k ⊗ V g ℓ ⊗ V h m ( r + 2) . By projecting AJ et (∆) we obtain a cohomology class κ ( f k , g ℓ , h m ) ∈ H 1 ( Q , V f k ⊗ V g ℓ ⊗ V h m ( r + 2)) , for each ( k , ℓ, m ) ∈ Σ bal .