p -adic Modular Forms and Arithmetic A conference in honor of Haruzo Hida’s 60th birthday Hida’s p -adic Rankin L -functions and syntomic regulators of Beilinson-Flach elements Henri Darmon UCLA, June 18, 2012
(Joint with Massimo Bertolini and Victor Rotger)
Also based on earlier work with Bertolini and Kartik Prasanna
Preliminaries Rankin L -series are attached to a pair f ∈ S k (Γ 1 ( N f ) , χ f ) , g ∈ S ℓ (Γ 1 ( N g ) , χ g ) of cusp forms, ∞ ∞ � a n ( f ) q n , � a n ( g ) q n . f = g = n =1 n =1 Hecke polynomials ( p ∤ N := lcm ( N f , N g )) x 2 − a p ( f ) x + χ f ( p ) p k − 1 = ( x − α p ( f ))( x − β p ( f )) . x 2 − a p ( g ) x + χ g ( p ) p ℓ − 1 = ( x − α p ( g ))( x − β p ( g )) .
Rankin L -series, definition Incomplete Rankin L -series: L N ( f ⊗ g , s ) − 1 � (1 − α p ( f ) α p ( g ) p − s )(1 − α p ( f ) β p ( g ) p − s ) = p ∤ N × (1 − β p ( f ) α p ( g ) p − s )(1 − β p ( f ) β p ( g ) p − s ) This definition, completed by a description of Euler factors at the “bad primes”, yields the Rankin L -series L ( f ⊗ g , s ) = L ( V f ⊗ V g , s ) , where V f , V g are the Deligne representations attached to f and g .
Rankin L -series, integral representation Assume for simplicity that k = ℓ = 2. Non-holomorphic Eisenstein series of weight 0: ′ � χ − 1 ( n ) y s | mz + n | − 2 s . E χ ( z , s ) = ( m , n ) ∈ N Z × Z Theorem (Shimura) Let χ := ( χ f χ g ) − 1 . Then L ( f ⊗ g , s ) = (4 π ) s � ¯ � f ( z ) , E χ ( z , s − 1) g ( z ) Γ 0 ( N ) . Γ( s ) This is proved using the Rankin-Selberg method .
Rankin L -series, properties The non-holomorphic Eisenstein series have analytic continuation to s ∈ C and satisfy a functional equation under s ↔ 1 − s . Shimura’s integral representation for L ( f ⊗ g , s ) leads to its analytic continuation, with a functional equation L ( f ⊗ g , s ) ↔ L ( f ⊗ g , 3 − s ) . Goal of Beilinson’s formula : Give a geometric interpretation for L ( f ⊗ g , s ) at the “near central point” s = 2. This geometric interpretation involves the higher Chow groups of X 0 ( N ) × X 0 ( N ).
Higher Chow groups Let S =smooth proper surface over a field K . Definition The Higher Chow group CH 2 ( S , 1) is the first homology of the Gersten complex ∂ � ⊕ Z ⊂ S K ( Z ) × div � ⊕ P ∈ S Z . K 2 ( K ( S )) So an element of CH 2 ( S , 1) is described by a formal linear combination of pairs ( Z j , u j ) where the Z j are curves in S , and u j is a rational function on Z j .
Beilinson-Flach elements These are distinguished elements in CH 2 ( S , 1) arising when 1 S = X 1 ( N ) × X 1 ( N ) is a product of modular curves; 2 Z = ∆ ≃ X 1 ( N ) is the diagonal ; 3 u ∈ C (∆) × is a modular unit. Lemma For all modular units u ∈ C (∆) × , there is an element of the form � � ∆ u = (∆ , u ) + λ i ( P j × X 1 ( N ) , u i ) + η j ( X 1 ( N ) × Q j , v j ) i j which belongs to CH 2 ( S , 1) ⊗ Q . It is called the Beilinson-Flach element associated to the pair (∆ , u ) .
Modular units Y 1 ( N ) / 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 − → Eis 2 (Γ 1 ( N ) , Q ) to the space of weight two Eisenstein series with coefficients in Q . Let u χ ∈ O × Y 1 ( N ) ⊗ Q χ be the modular unit characterised by dlog u χ = E 2 ,χ , ∞ E 2 ,χ ( z ) = 2 − 1 L ( χ, − 1) + � σ χ ( n ) q n , � σ χ ( n ) = χ ( d ) d . n =1 d | n
Complex regulators The complex regulator is the map → (Fil 1 H 2 reg C : CH 2 ( S , 1) − dR ( S / C )) ∨ defined by 1 � reg C (( Z , u ))( ω ) = Z ′ ω log | u | , 2 π i where ω is a smooth two-form on S whose associated class in H 2 dR ( S / C ) belongs to Fil 1 ; Z ′ =locus in Z where u is regular.
Beilinson’s formula Theorem (Beilinson) For cusp forms f and g of weight 2 and characters χ f and χ g , L ( f ⊗ g , 2) = C χ × reg C (∆ u χ )(¯ ω f ∧ ω g ) , where C χ = 16 π 3 N − 2 τ ( χ − 1 ) , χ = ( χ f χ g ) − 1 .
A p -adic Beilinson formula? Such a formula should relate: 1 The value at s = 2 of certain p -adic L -series attached to f and g ; 2 The images of Beilinson-Flach elements under certain p-adic syntomic regulators , in the spirit of Coleman-de Shalit, Besser.
Hida’s p -adic Rankin L -series To define L p ( f ⊗ g , s ), the obvious approach is to interpolate the values L ( f ⊗ g , χ, j ) , χ a Dirichlet character , j ∈ Z . Difficulty : none of these ( χ, j ) are critical in the sense of Deligne. Hida’s solution : “enlarge” the domain of definition of L p ( f , g , s ) by allowing f and g to vary in p-adic families .
Hida families Iwasawa algebra : Λ = Z p [[1 + p Z p ]] ≃ Z p [[ T ]]: 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 f , Ω f , f ) , where 1 Λ f is a finite flat extension of Λ; 2 Ω f ⊂ X f := hom(Λ f , C p ) is a non-empty open subset (for the p -adic topology); n a n q n ∈ Λ f [[ q ]] is a formal q -series, such that 3 f = � n x ( a n ) q n is the q series of the ordinary f ( x ) := � p-stabilisation f ( p ) of a normalised eigenform, denoted f x , of x weight κ ( x ) on Γ 1 ( N ), for all x ∈ Ω f , cl := Ω f ∩ X f , cl .
Hida’s theorem f = normalised eigenform of weight k ≥ 1 on Γ 1 ( N ). p ∤ N an ordinary prime for f (i.e., a p ( f ) is a p -adic unit). Theorem (Hida) There exists a Hida family (Λ f , Ω f , f ) and a classical point x 0 ∈ Ω f , cl satisfying κ ( x 0 ) = k , f x 0 = f . As x varies over Ω f , cl , the specialisations f x give rise to a “ p -adically coherent” collection of classical newforms on Γ 1 ( N ), and one can hope to construct p -adic L -functions by interpolating classical special values attached to these eigenforms.
Hida’s p -adic Rankin L -functions They should interpolate critical values of the form L ( f x ⊗ g y , j ) Ω( f x , g y , j ) ∈ ¯ Q , ( x , y , j ) ∈ Ω f , cl × Ω g , cl × Z . Proposition The special value L ( f x ⊗ g y , j ) is critical if and only if either: κ ( y ) ≤ j ≤ κ ( x ) − 1 ; then Ω( f x , g y , j ) = ∗� f x , f x � . κ ( x ) ≤ j ≤ κ ( y ) − 1 ; then Ω( f x , g y , j ) = ∗� g y , g y � . Let Σ f , Σ g ⊂ Ω f × Ω g × Ω be the two sets of critical points. Note that they are both dense in the p -adic domain.
Hida’s p -adic Rankin L -functions Theorem (Hida) There are two ( a priori quite distinct ) p-adic L-functions, L f L g p ( f ⊗ g ) , p ( f ⊗ g ) : Ω f × Ω g × Ω − → C p , interpolating the algebraic parts of L ( f x ⊗ g y , j ) for ( x , y , j ) belonging to Σ f and Σ g respectively.
� � � � p -adic regulators reg et et (¯ CH 2 ( S / Z , 1) H 1 f ( Q , H 2 S , Q p )(2)) � H 1 et (¯ CH 2 ( S / Z p , 1) f ( Q p , H 2 S , Q p )(2)) reg et log p Fil 1 H 2 dR ( S / Q p ) ∨ The dotted arrow is called the p-adic regulator and denoted reg p .
Syntomic regulators Coleman-de Shalit, Besser : A direct, p -adic analytic description of the p -adic regulator in terms of Coleman’s theory of p -adic integration.
The p -adic Beilinson formula: the set-up f = Hida family of tame level N specialising to the weight two cusp form f ∈ S 2 (Γ 0 ( N ) , χ f ) at x 0 ∈ Ω f . g = Hida family of tame level N specialising to the weight two cusp form g ∈ S 2 (Γ 0 ( N ) , χ g ) at y 0 ∈ Ω g . χ = ( χ f χ g ) − 1 . dR ( X 0 ( N ) / C p ) f which is in the unit root η ur = unique class in H 1 f subspace for Frobenius and satisfies � ω f , η ur f � = 1.
The p -adic Beilinson formula Theorem (Bertolini, Rotger, D) p ( f , g )( x 0 , y 0 , 2) = E ( f , g , 2) L f E ( f ) E ∗ ( f ) × reg p (∆ u χ )( η ur f ∧ ω g ) , p ( f , g )( x 0 , y 0 , 2) = E ( g , f , 2) E ( g ) E ∗ ( g ) × reg p (∆ u χ )( ω f ∧ η ur L g g ) , where E ( f , g , 2) = (1 − β p ( f ) α p ( g ) p − 2 (1 − β p ( f ) β p ( g ) p − 2 ) × (1 − β p ( f ) α p ( g ) χ ( p ) p − 1 )(1 − β p ( f ) β p ( g ) χ ( p ) p − 1 ) E ( f ) = 1 − β p ( f ) 2 χ − 1 ( p ) p − 2 , E ∗ ( f ) = 1 − β p ( f ) 2 χ − 1 ( p ) p − 1 . f f
Arithmetic applications: Dasgupta’s formula In his work on the L -invariant for the symmetric square, Dasgupta is led to study L Hida ( f , f ) when f = g , and its restriction p L Hida ( f , f )( x , x , j ) to the diagonal in Ω f × Ω f . p This restriction has no critical values . The “Artin formalism” for p -adic L -functions suggests that it should factor into a product of 1 the Coates-Schmidt p -adic L -function L CS p ( Sym 2 ( f ))( x , j ), which does have critical points; 2 the Kubota-Leopoldt p -adic L -function L KL p ( χ f , j + 1 − κ ( x )).
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