Special values of triple-product p -adic L -functions and non-crystalline diagonal classes
aa r X i v : . [ m a t h . N T ] D ec SPECIAL VALUES OF TRIPLE-PRODUCT p -ADIC L-FUNCTIONS ANDNON-CRYSTALLINE DIAGONAL CLASSES FRANCESCA GATTI, XAVIER GUITART, MARC MASDEU, AND VICTOR ROTGER
Abstract.
The main purpose of this note is to understand the arithmetic encoded in the specialvalue of the p -adic L -function L gp ( f , g , h ) associated to a triple of modular forms ( f, g, h ) of weights(2 , , L -function L ( f ⊗ g ⊗ h, s ) –which typically has sign +1–does not vanish at its central critical point s = 1. When f corresponds to an elliptic curve E/ Q and the classical L -function vanishes, the Elliptic Stark Conjecture of Darmon–Lauder–Rotgerpredicts that L gp ( f , g , h )(2 , ,
1) is either 0 (when the order of vanishing of the complex L -functionis >
2) or related to logarithms of global points on E and a certain Gross–Stark unit associatedto g . We complete the picture proposed by the Elliptic Stark Conjecture by providing a formulafor the value L gp ( f , g , h )(2 , ,
1) in the case where L ( f ⊗ g ⊗ h, = 0. Contents
1. Introduction 12. The Selmer group of f ⊗ g ⊗ h V p -adic L -function in rank 0 114. The case of theta series of an imaginary quadratic field K where p splits 145. Numerical computations 165.1. Dihedral case 165.2. Exotic image case 18References 191. Introduction
Let E be an elliptic curve defined over Q and let f ∈ S ( N f ) be the newform attached to E . Let g ∈ S ( N g , χ ) L , h ∈ S ( N h , ¯ χ ) L Date : December 18, 2019.This project has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement No 682152). The authors were also partiallysupported by projects MTM2015-66716-P and MTM2015-63829-P. p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 2 be two cuspforms of weight one, inverse nebentype characters and with Fourier coefficients containedin a number field L . Let ρ g and ρ h be the Artin representations attached to g and h . The tensorproduct ρ g ⊗ ρ h is a self-dual Artin representation of dimension 4 of the form ρ := ρ g ⊗ ρ h : Gal( H/ Q ) ֒ → Aut( V g ⊗ V h ) ∼ = GL ( L ) , where H/ Q is a finite extension.In this setting, the complex L -function L ( E ⊗ ρ, s ) attached to the (Tate module V p ( E ) of the)elliptic curve E twisted by the Artin representation ρ coincides with the Garrett–Rankin L -function L ( f ⊗ g ⊗ h, s ) attached to the triple ( f, g, h ) of modular forms. By multiplying this L -function byan appropriate archimedean factor L ∞ ( f ⊗ g ⊗ h, s ) one obtains an entire function Λ( f ⊗ g ⊗ h, s )which satisfies a functional equation of the form(1.1) Λ( f ⊗ g ⊗ h, s ) = ǫ · Λ( f ⊗ g ⊗ h, − s ) , where ǫ ∈ {± } . Moreover, L ∞ ( f ⊗ g ⊗ h, s ) does not have zeros nor poles at s = 1.Denote N g and N h the level of g and h respectively. The sign can be written as a product oflocal factors ǫ = Q v ǫ v where v runs over the places of Q , and ǫ v = +1 if v is a finite prime whichdoes not divide lcm( N f , N g , N h ) or if v = ∞ . We will work under the following assumption Assumption 1.1. ǫ v = +1 for all v. Assumption 1.1 holds most of the time: this is the case for instance if the greatest commondivisor of the levels of f, g and h is 1.Fix an odd prime number p such that p ∤ N f N g N h , and denote by α g , β g the eigenvalues for the action of the Frobenius element at p acting on V g . Weuse the analogous notation for h , and we assume α g = β g , and α h = β h . Fix once and for all completions H p , L p of the number fields H, L at primes above p .Choose an ordinary p -stabilisation of g , namely g α ( z ) := g ( z ) − β g g ( pz ) and define analogously h α . Let f , g , h be Hida families passing through the unique ordinary p -stabilisation of f and g α and h α respectively, and consider the Garret–Hida p -adic L -function L gp ( f , g , h )of [DR14] associated to the specific choice of test vectors (˘ f , ˘ g , ˘ h ) of [Hsi17, Chap. 3]. This p -adic L -function interpolates the square-roots of the central values of the classical L -function L ( ˘ f k ⊗ ˘ g ℓ ⊗ ˘ h m , s ) attached to the specializations of the Hida families at classical points of weights k, ℓ, m with k, ℓ, m ≥ ℓ ≥ k + m . Notice that the point (2 , , f, g, h ), lies outside the region of classical interpolation for L gp ( f , g , h ). Weare interested in studying the value L gp ( f , g , h )(2 , , Assumption 1.2. L ( E ⊗ ρ, = 0 and Sel p ( E ⊗ ρ ) = 0 . Here Sel p ( E ⊗ ρ ) denotes the Bloch–Kato Selmer group attached to the representation V := V p ( E ) ⊗ V g ⊗ V h . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 3 Under Assumption 1.1, the sign ǫ of the functional equation (1.1) is +1, and thus the orderof vanishing of L ( E ⊗ ρ, s ) at s = 1 is even. One hence expects that L ( E ⊗ ρ,
1) is genericallynonzero. If this L -value is nonzero, by [DR17] we know that the ρ -isotypical component E ( H ) ρ :=Hom G Q ( V g ⊗ V h , E ( H ) ⊗ L ) of the Mordell-Weil group E ( H ) is trivial. By the Shafarefich–Tateconjecture one also expects the Selmer group Sel p ( E ⊗ ρ ) to be trivial, although this conjecture iswidely open. It is also worth noting that the value L gp ( f , g , h )(2 , ,
1) in the setting in which thecomplex L -function L ( E ⊗ ρ, s ) vanishes at s = 1 has been analyzed in [DLR15], where the authorsgive a conjectural formula for this p -adic value as a 2 × p -adic logarithms of globalpoints.Under our running assumption 1.2 one can not expect a similar formula for the above p -adic L -value, as no global points are naturally present in this scenario. The main result of this paper consists in an explicit formula for the value L gp ( f , g , h )(2 , ,
1) which involves the algebraic partof the classical L -value L ( E ⊗ ρ,
1) and the logarithm of a canonical non-crystalline class along acertain crystalline direction .In § p -Selmer group Sel ( p ) ( E ⊗ ρ ) under Assumption 1.2. More precisely, the projection to thesingular quotient gives an isomorphism(1.2) ∂ p : Sel ( p ) ( E ⊗ ρ ) ∼ = −→ H s ( Q p , V ) . Let V αg , V βg , with basis v αg , v βg respectively, be the eigenspaces of V g for the action of Frob p witheigenvalues α g , β g , and use the analogous notation for V h . The G Q p -representation V decomposesas a direct sum as V = V αα ⊕ V αβ ⊕ V βα ⊕ V ββ , where V αα := V p E ⊗ V αg ⊗ V αh and similarly for the other pieces. It induces the decomposition(1.3) H s ( Q p , V ) = H s ( Q p , V αα ) ⊕ H s ( Q p , V αβ ) ⊕ H s ( Q p , V βα ) ⊕ H s ( Q p , V ββ ) , and the Bloch–Kato dual exponential gives isomorphismsexp ∗ αα : H s ( Q p , V αα ) ∼ = −→ L p and similarly for the other pieces of the decomposition (1.3). Combining it with (1.2), we get abasis ξ αα , ξ αβ , ξ βα , ξ ββ for Sel ( p ) ( E ⊗ ρ ) characterised by the fact that ∂ p ξ αα ∈ H s ( Q p , V αα ) and exp ∗ αα ∂ p ξ αα = 1 , and similarly for ξ αβ , ξ βα , ξ ββ .The G Q p -cohomology of V and its submodule of crystalline classes H f ( Q p , V ) ⊆ H ( Q p , V ) alsohave decompositions analogous to (1.3). Moreover, if π αβ : H ( Q p , V ) −→ H ( Q p , V αβ )denotes the projection, then π αβ ξ ββ lies in H f ( Q p , V αβ ). Finally, we can write π αβ ξ ββ = R βα ⊗ v αg ⊗ v βh ∈ ( E ( H p ) ⊗ V αg ⊗ V βh ) G Q p ∼ = H f ( Q p , V αβ )where R βα ∈ E ( H p ) is a local point on which Frob p acts as multiplication by β g α h .We can finally state the main result of the paper. RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 4 Theorem (cf Theorem 3.2) . Under Assumptions 2.1 and 1.2, (1.4) L gp ( f , g , h )(2 , ,
1) = A · E π h f, f i × log p ( R βα ) L g α × p L ( E ⊗ ρ, , where A ∈ Q × is an explicit number, E ∈ L p is a product of Euler factors, h f, f i denotes thePetersson norm of f , L g α ∈ H p is an element on which Frob p acts as multiplication by β g α g andwhich only depends on g α , and log p : E ( H p ) → H p denotes the p -adic logarithm. We refer to Theorem 3.2 for a more precise statement of the result and of the objects appearingin (1.4). In particular, the element L g α is expected to be related to a so-called Gross–Stark unitattached to g α , as conjectured in [DR16, Conjecture 2.1].Under the additional assumption that g is not the theta series of a Hecke character of a realquadratic field in which p splits, the value L gp ( f , g , h )(2 , ,
1) can be recast in a more explicit wayin terms of p -adic iterated integrals , as explained in the introduction of [DLR15]. The numericalcomputations we offer in § g and h aretheta series of the same imaginary quadratic field in which p splits. The following theorem is statedas Theorem 4.1 in the text. Theorem (cf Theorem 4.1) . Let K be an imaginary quadratic field in which p is split, and let ψ g (resp. ψ h ) be a finite order Hecke character of K of conductor c g (resp. of conductor c h ) . Denoteby g and h the theta series attached to ψ g and ψ h , respectively. Suppose that gcd( N f , c g , c h ) = 1 and that the Nebentype characters of g and h are inverses to each other. If L ( E, ρ g ⊗ ρ h , = 0 then L gp ( f , g , h )(2 , ,
1) = 0 . The Selmer group of f ⊗ g ⊗ h We begin this section by collecting some standard facts on Selmer groups of p -adic Galois rep-resentations that we will use. Then we introduce the Galois representation attached to the tripleof modular forms f , g , and h of weights 2, 1, 1, and we study the corresponding Selmer groups. Inparticular, the structure of the relaxed Selmer group will be key in proving the main theorem ofSection 3.2.1. Selmer groups.
Let V be a Q p [ G Q ]-module and let B cris be Fontaine’s p -adic crystallineperiod ring. For each prime number ℓ , denote(2.1) H f ( Q ℓ , V ) := ( H ( Q ℓ , V ) := H ( Q ur ℓ / Q ℓ , V I ℓ ) ℓ = p ker (cid:16) H ( Q p , V ) → H ( Q p , V ⊗ Q p B cris ) (cid:17) ℓ = p, and H s ( Q ℓ , V ) := H ( Q ℓ , V ) / H f ( Q ℓ , V ) . The Bloch–Kato Selmer group of V isSel p ( Q , V ) := { x ∈ H ( Q , V ) | res ℓ ( x ) ∈ H f ( Q ℓ , V ) for all ℓ } , where res ℓ : H ( Q , V ) → H ( Q ℓ , V ) denotes the restriction map in Galois cohomology.For each prime ℓ , we denote by ∂ ℓ the composition ∂ ℓ : H ( Q , V ) res ℓ −→ H ( Q ℓ , V ) −→ H s ( Q ℓ , V ) , RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 5 where the second map is the natural quotient map.The relaxed Selmer group is defined asSel ( p ) ( Q , V ) := { x ∈ H ( Q , V ) | res ℓ ( x ) ∈ H f ( Q ℓ , V ) for all ℓ = p } ⊇ Sel p ( Q , V ) . Let V ∗ := Hom Q p ( V, Q p (1)) be the Kummer dual of V . One can define a Selmer groupSel p, ∗ ( Q , V ∗ ) for V ∗ which is dual to (2.1) with respect to the local Tate pairings h , i ℓ : H ( Q ℓ , V ) × H ( Q ℓ , V ∗ ) −→ Q p . (2.2)For each ℓ , define H f, ∗ ( Q ℓ , V ∗ ) to be the orthogonal complement of H f ( Q ℓ , V ) with respect to (2.2);the Selmer group attached to V ∗ is thenSel p, ∗ ( Q , V ∗ ) := { x ∈ H ( Q , V ∗ ) | res ℓ ( x ) ∈ H f, ∗ ( Q ℓ , V ∗ ) for all ℓ } . Finally, the strict Selmer group of V ∗ is the subspace of Sel p, ∗ ( Q , V ∗ ) defined asSel [ p ] , ∗ ( Q , V ∗ ) := { x ∈ H ( Q , V ∗ ) | res ℓ ( x ) ∈ H f, ∗ ( Q ℓ , V ∗ ) for all ℓ and res p ( x ) = 0 } . By Poitou–Tate duality (see, for example, [MR04, Theorem 2.3.4]) there is an exact sequence(2.3) 0 → Sel p ( Q , V ) → Sel ( p ) ( Q , V ) → H s ( Q p , V ) → Sel p, ∗ ( Q , V ∗ ) ∨ → Sel [ p ] , ∗ ( Q , V ∗ ) ∨ , where ∨ stands for the Q p -dual.2.2. Representations attached to modular forms.
In this section we review the main featuresof the representations, both p -adic and Λ-adic, attached to modular forms in the lines of [DR16, § f ∈ S ( N f ) be a weight two normalized eigenform of level N f , trivial nebentype characterand rational Fourier coefficients a n ( f ). Denote by E the elliptic curve over Q of conductor N f associated to f by the Eichler–Shimura construction.Let also g ∈ S ( N g , χ ) and h ∈ S ( N h , ¯ χ )be two normalized newforms of weight one, levels N g and N h , and nebentype characters χ and ¯ χ respectively. Denote by K g and K h their fields of Fourier coefficients, and put L := K g · K h thecompositum of these fields.From now on, we fix a rational prime p , and we assume the following hypothesis. Assumption 2.1.
The prime p does not divide N f N g N h . Since we will be interested in putting f in a Hida family, we assume moreover that f is ordinaryat p ; that is to say, that p ∤ a p ( f ).We denote the 2-dimensional p -adic representations attached to f by V f . Since f correspondsto the curve E , the representation V f is given by the rational Tate module V p ( E ) = T p ( E ) ⊗ Q p .Denote by α f , β f the roots of the Hecke polynomial X − a p ( f ) X + p . Since f is ordinary at p ,one of these roots, say α f , is a p -adic unit; also, the restriction of V f to a decomposition group G Q p ⊂ G Q admits a filtration of Q p [ G Q p ]-modules0 → V + f −→ V f −→ V − f → Q p V + f = dim Q p V − f = 1;(2) the group G Q p acts on the quotient V − f via ψ f , where ψ f : G Q p → Z × p is the unramifiedcharacter that maps an arithmetic Frobenius Frob p to α f . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 6 (3) the group G Q p acts on V + f via the character χ cycl ψ − f (here χ cycl is the p -adic cyclotomiccharacter).There are Artin representations associated to g and h . Without loss of generality we can assumethat they are defined over L , and that they factor through the same finite extension H of Q . Thatis to say, they are of the form ρ g : Gal( H/ Q ) −→ Aut( V ❛ g ) ∼ = GL ( L ) , ρ h : Gal( H/ Q ) −→ Aut( V ❛ h ) ∼ = GL ( L )for certain 2-dimensional L -vector spaces V ❛ g and V ❛ h .Fix once and for all a prime p of H and a prime P of L above p . Denote the correspondingcompletions by H p := H p and L p := L P . There are also p -adic Galois representations associatedto g and h , that we will denote by V g and V h . There are non-canonical isomorphisms j g : V ❛ g ⊗ L L p ∼ = −→ V g and j h : V ❛ h ⊗ L L p ∼ = −→ V h . (2.4)Since p ∤ N g N h the representations V g and V h are unramified at p . We assume from now on thatFrob p acts on V g and V h with distinct eigenvalues. Let α g , β g be the eigenvalues for the action ofFrob p on V g and let V αg , V βg be the corresponding eigenspaces. We will use the analogous notations α h , β h , V αh , and V βh for h .Denote by g α the p -stabilisation of g such that U p ( g α ) = α g g α . The theory of Hida familiesensures the existence of a Hida family g passing through g α . This can be regarded as a power series g ∈ Λ g [[ q ]], where Λ g is a finite flat extension of the Iwasawa algebra Λ := Z p [[ T ]], with the propertythat, if we denote by y g : Λ g → L p the weight corresponding to g , then y g ( g ) = g α . There is a locallyfree Λ g -module V g and a Λ-adic representation ρ g : G Q → GL( V g ) ∼ = GL (Λ g ) that interpolatesthe p -adic representations associated to the specializations of g . As a G Q p -representation, V g isequipped with a filtration of Λ g [ G Q p ]-modules(2.5) 0 → V + g −→ V g −→ V − g → , where V + g and V − g are locally free of rank one and the action of G Q p on V − g is unramified, with Frob p acting as multiplication by the p -th Fourier coefficient of g . There is a perfect Galois equivariantpairing h , i : V − g × V + g −→ Λ g (det( ρ g )) . (2.6)For a crystalline Q p [ G Q p ]-module W , denote D ( W ) := ( W ⊗ B cris ) G Q p . Recall that, if W isunramified, then D ( W ) ∼ = ( W ⊗ ˆ Q ur p ) G Q p , where ˆ Q ur p is the p -adic completion of the maximal unramified extension of Q p . Denote by ω g ∈ D ( V − g ) the canonical period associated to g constructed by Ohta [Oht95].By specializing via y g , we obtain the L p -vector space y g ( V g ) := V g ⊗ Λ g ,y g L p , which can beidentified with V g . Using the functoriality of D and the identification y g ( V + g ) = V βg , y g ( V − g ) = V αg we obtain a pairing(2.7) h , i : D ( V αg ) × D ( V βg ) −→ D ( L p ( χ )) = ( H p ⊗ L p ( χ )) G Q p . Define ω g := y g ( ω g ) ∈ D ( V αg ) RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 7 and let η g ∈ D ( V βg )be the element characterized by the equality(2.8) h ω g , η g i = g ( χ ) ⊗ ∈ D ( L p ( χ )) , where g ( χ ) denotes the Gauss sum of χ viewed as an element of H p . We define similarly ω h ∈ D ( V αh )and η h ∈ D ( V βh ).Using the isomorphisms (2.4) we can define an L structure on V g by V Lg := j g ( V ❛ g ). Let v αg (resp. v βg ) be an L -basis of V Lg ∩ V αg (resp. of V βg ). DefineΩ g ∈ H /α g p , Θ g ∈ H /β g p to be the elements such that(2.9) Ω g ⊗ v αg = ω g ∈ D ( V αg ) , Θ g ⊗ v βg = η g ∈ D ( V βg ) . Let V := V f ⊗ V gh be the p -adic representation given by the tensor product V f ⊗ V g ⊗ V h . Since the product of thenebentype characters of f , g , and h is trivial we have that V ∗ ∼ = V . We next study the structureof several Selmer groups associated to V .2.3. Selmer groups of V . Put V ❛ gh := V ❛ g ⊗ L V ❛ h and denote by ρ the representation afforded bythis space: ρ : Gal( H/ Q ) −→ Aut( V ❛ gh ) . Put E ( H ) L := E ( H ) ⊗ Z L and denote by E ( H ) ρ the ρ -isotypical component of the Mordell–Weilgroup: E ( H ) ρ := Hom Gal( H/ Q ) ( V ❛ gh , E ( H ) L ) . Lemma 2.2.
There are isomorphisms H ( Q , V ) ∼ = (H ( H, V f ) ⊗ V gh ) Gal( H/ Q ) ∼ = Hom Gal( H/ Q ) ( V gh , H ( H, V f ));(2.10) H ( Q p , V ) ∼ = (H ( H p , V f ) ⊗ V gh ) Gal( H p / Q p ) ∼ = Hom Gal( H p / Q p ) ( V gh , H ( H p , V f )) . (2.11) Proof.
We prove only (2.11), and (2.10) is proven similarly. By the inflation-restriction exactsequence we have the exact sequence0 → H (Gal( H p / Q p ) , V G Hp ) → H ( Q p , V ) → H ( H p , V ) Gal( H p / Q p ) → H (Gal( H p / Q p ) , V G Hp ) . Since H (Gal( H p / Q p ) , V G Hp ) = H (Gal( H p / Q p ) , V G Hp ) = 0, the restriction to G H p gives an iso-morphism H ( Q p , V ) −→ H ( H p , V ) Gal( H p / Q p ) . Composing it with the identificationsH ( H p , V ) Gal( H p / Q p ) = H ( H p , V f ⊗ V gh ) Gal( H p / Q p ) = (H ( H p , V f ) ⊗ V gh ) Gal( H p / Q p ) , we get the first isomorphism of (2.11). Finally, the second isomorphism follows from the relationbetween Hom and tensor and from the selfduality V ∨ gh ∼ = V gh . (cid:3) RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 8 Let E ( H ) V gh L := Hom Gal( H/ Q ) ( V gh , E ( H ) L ). The Kummer homomorphism E ( H ) L −→ H ( H, V f )induces a homomorphism δ : E ( H ) V gh L −→ Hom
Gal( H/ Q ) ( V gh , H ( H, V f )) ∼ = H ( Q , V ) , which using (2.10) can be seen as an isomorphism δ : E ( H ) V gh L −→ H ( Q , V ) . For △ , ♥ ∈ { α, β } , denote V △♥ gh := V △ g ⊗ V ♥ h and V △♥ := V f ⊗ V △ g ⊗ V ♥ h . Specializing (2.5) via y g we obtain0 −→ V βg −→ V g −→ V αg −→ . For △ 6 = ♥ the pairing (2.6) and its analog for h induce perfect pairings h , i : V △△ gh × V ♥♥ gh −→ L p , h , i : V △♥ g × V ♥△ g −→ L p . (2.12)The identifications V △△ gh ∼ = Hom L p [ G Q p ] ( V ♥♥ gh , L p ) and V △♥ gh ∼ = Hom L p [ G Q p ] ( V ♥△ gh , L p ) , (2.13)together with (2.11) give the following isomorphisms:H ( Q p , V △△ ) ∼ = (H ( H p , V f ) ⊗ V △△ gh ) Gal( H p / Q p ) ∼ = Hom Gal( H p / Q p ) ( V ♥♥ gh , H ( H p , V f ));(2.14) H ( Q p , V △♥ ) ∼ = (H ( H p , V f ) ⊗ V △♥ gh ) Gal( H p / Q p ) ∼ = Hom Gal( H p / Q p ) ( V ♥△ gh , H ( H p , V f )) . (2.15)It follows from [DR19, Lemma 4.1] that the submodule H f ( Q p , V f ⊗ V △♥ gh ) and the singularquotient H s ( Q p , V f ⊗ V △♥ gh ) can be written in terms of the filtration of V f as follows:H s ( Q p , V f ⊗ V △♥ gh ) = H ( Q p , V − f ⊗ V △♥ gh ) ∼ = ( V ♥△ gh ⊗ H s ( H p , V f )) Gal( H p / Q p ) ;(2.16)H f ( Q p , V f ⊗ V △♥ gh ) = ker(H ( Q p , V f ⊗ V △♥ gh ) → H ( I p , V − f ⊗ V △♥ gh )) = H ( Q p , V + f ⊗ V △♥ gh ) . (2.17) Lemma 2.3.
For △ , ♥ ∈ { α, β } , △ 6 = ♥ , there are isomorphisms δ p : E ( H p ) V gh L p −→ H f ( Q p , V ); δ △♥ p : E ( H p ) V △♥ gh L p −→ H f ( Q p , V ♥△ ); δ △△ p : E ( H p ) V △△ gh L p −→ H f ( Q p , V ♥♥ ) . Proof.
We prove the existence of the isomorphism δ △♥ , the others are similar. By Kummer theory,there is an injective morphism E ( H p ) L p −→ H ( H p , V f ) , which is an isomorphism on its image H f ( H p , V f ) ∼ = H ( H p , V + f ). It induces an isomorphism δ △♥ p : E ( H p ) V △♥ gh L p −→ Hom
Gal( H p / Q p ) ( V △♥ gh , H ( H p , V + f )) . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 9 Using the isomorphisms (2.13) we obtainHom
Gal( H p / Q p ) ( V △♥ gh , H ( H p , V + f )) ∼ = −→ (H ( H p , V + f ) ⊗ V ♥△ gh ) Gal( H p / Q p ) . Arguing as in the proof of Lemma 2.2, we get the isomorphisms(H ( H p , V + f ) ⊗ V ♥△ gh ) Gal( H p / Q p ) ∼ = H ( H p , V + f ⊗ V ♥△ gh ) Gal( H p / Q p ) ∼ = H ( Q p , V + f ⊗ V ♥△ gh ) ∼ = H f ( Q p , V ♥△ ) . (cid:3) From now on we will make the following assumption on the Selmer group of V . Assumption 2.4.
Sel p ( Q , V ) = 0.Under this assumption one can identify the relaxed Selmer group with the singular quotient. Lemma 2.5.
Under Assumptions 2.4 and 2.1 the natural map ∂ p : Sel ( p ) ( Q , V ) −→ H s ( Q p , V ) is an isomorphism. In particular, there is an isomorphism Sel ( p ) ( Q , V ) ∼ = H s ( Q p , V αα ) ⊕ H s ( Q p , V αβ ) ⊕ H s ( Q p , V βα ) ⊕ H s ( Q p , V ββ )(2.18) Proof.
Since the representation V is self-dual there is an isomorphism Sel p, ∗ ( V ∗ ) ∼ = Sel p ( V ) , (see,for example, [BK90] and [Bel, Theorem 2.1]). Then the lemma follows immediately from the exactsequence (2.3). (cid:3) In the next subsection we will describe the spaces in the right hand side of (2.18) in terms ofdual exponential maps.2.4.
Bloch–Kato logarithms and exponentials.
The G Q p -representations of the form V + f ⊗ V △♥ gh are one dimensional and, therefore, given by characters. Indeed, G Q p acts on V + f as χ cycl ψ − f , andit acts as ψ g (resp. ψ − g ) on V αg (resp. V βg ) and as ψ h (resp. ψ − h ) on V αh (resp. V βh ). Therefore wehave that V + f ⊗ V ααgh = L p ( χ cycl ψ − f ψ g ψ h ) , V + f ⊗ V αβgh = L p ( χ cycl ψ − f ψ g ψ − h ) ,V + f ⊗ V βαgh = L p ( χ cycl ψ − f ψ − g ψ h ) , V + f ⊗ V ββgh = L p ( χ cycl ψ − f ψ − g ψ − h ) . In particular V + f ⊗ V △♥ gh is of the form L p ( ψχ cycl ) for some nontrivial unramified character ψ . By(2.17) we have that H f ( Q p , V △♥ ) ∼ = H ( Q p , V + f ⊗ V △♥ gh ), and the Bloch–Kato logarithm gives anisomorphism (cf. [DR19, Example 1.6 (a)]):(2.19) log △♥ : H f ( Q p , V △♥ ) −→ D ( V + f ⊗ V △♥ gh ) = D ( L p ( ψχ cycl )) . For ( △ , ♥ ) = ( α, α ), the pairings 2.7 and the analogous pairings for f and h give rise to a pairing(2.20) h , i : V + f ⊗ V αg ⊗ V αh × V − f ( − ⊗ V βg ⊗ V βh −→ L p which induces(2.21) h , i : D ( V + f ⊗ V αg ⊗ V αh ) × D ( V − f ( − ⊗ V βg ⊗ V βh ) −→ D ( L p ) = L p . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 10 Denote by ˜ ω f the differential form on X ( N f ) corresponding to f . It can be naturally viewed asan element of the de Rham cohomology group H ( X ( N f ) / Q p ). The comparison isomorphismsof p -adic Hodge theory provide a natural map H ( X ( N f ) / Q p )(1) −→ D ( V − f )and therefore ˜ ω f gives rise to an element ω f ∈ D ( V − f ( − ω f ⊗ η g ⊗ η h gives then an isomorphism(2.22) h· , ω f ⊗ η g ⊗ η h i : D ( L p ( ψχ cycl )) = D ( V + f ⊗ V αg ⊗ V αh ) −→ L p . There are similar pairings and isomorphisms for the remaining pairs ( △ , ♥ ). We still denote(2.23) log △♥ : H f ( Q p , V f ⊗ V △♥ gh ) −→ L p the map obtained by composing (2.19) with (2.22). Remark . The logarithm maps of (2.23) are related to the usual p -adic logarithm on E as follows.The differential ω f gives rise to an invariant differential on E , and we denote bylog f,p : E ( H p ) −→ H p the corresponding formal group logarithm on E . The map log αβ coincides with the inverse of theisomorphism of Lemma 2.3 E ( H p ) V βαgh L p ∼ = ( E ( H p ) β g α h ⊗ V αβgh ) G Q p composed with the maps( E ( H p ) β g α h ⊗ V αβgh ) G Q p −→ ( H β g α h p ⊗ V αβgh ) G Q p = D ( V αβgh ) −→ L p x ⊗ v αg v βh log f,p ( x ) ⊗ v αg v βh y y, η g ω h i . Analogous equalities hold for the other maps log △♥ .A similar discussion can be applied to the representations of the form V − f ⊗ V △♥ gh . In this casewe have the following isomorphisms of 1-dimensional representations: V − f ⊗ V ααgh = L p ( ψ f ψ g ψ h ) , V − f ⊗ V αβgh = L p ( ψ f ψ g ψ − h ¯ χ ) ,V − f ⊗ V βαgh = L p ( ψ f ψ − g ψ h χ ) , V − f ⊗ V ββgh = L p ( ψ f ψ − g ψ − h ) . Therefore, V − f ⊗ V △♥ gh is isomorphic to a representation of the form L p ( ψ ) for some unramified andnontrivial character ψ . By (2.16) there is an identificationH s ( Q p , V △♥ ) = H ( Q p , L p ( ψ )) , and by [DR17, Example 1.8 (b)] the dual exponential gives isomorphisms(2.24) exp ∗△♥ : H s ( Q p , V △♥ ) −→ D ( L p ( ψ )) ∼ = L p , where the last isomorphism is induced by pairing with the appropriate class of D ( L p ( ψ − )) = D ( V + f ( − ⊗ V ♥ g ⊗ V △ h ) similarly as in (2.22). Arguing as in Remark 2.6, letexp ∗ f,p : H s ( H p , V f ) −→ H p RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 11 denote the dual exponential on H s ( H p , V f ). Then exp ∗ ββ can be identified with the composition(H s ( H p , V f ) α g α h ⊗ V ββgh ) G Q p −→ ( H α g α h p ⊗ V ββgh ) G Q p = D ( V ββgh ) −→ L p x ⊗ v βg v βh exp ∗ f,p ( x ) ⊗ v βg v βh y y, ω g ω h i , (2.25)after taking into account the identificationH s ( Q p , V ββ ) ∼ = (H s ( H p , V f ) α g α h ⊗ V ββgh ) G Q p . Analogous formulas hold for the dual exponentials exp △♥ on the remaining components.To sum up the discussion of this subsection, we conclude that the relaxed Selmer group of V admits a basis adapted to decomposition (2.18) with respect to the dual exponential maps. Proposition 2.7.
Under Assumptions 2.4 and 2.1,
Sel ( p ) ( V ) has a basis { ξ αα , ξ αβ , ξ βα , ξ ββ } (2.26) characterized by the fact that there exist elements Ψ ββ ∈ H s ( H p , V f ) β g β h , Ψ βα ∈ H s ( H p , V f ) β g α h , Ψ αβ ∈ H s ( H p , V f ) α g β h , Ψ αα ∈ H s ( H p , V f ) α g α h such that ∂ p ξ αα = (Ψ ββ ⊗ v αg v αh , , , , ∂ p ξ αβ = (0 , Ψ βα ⊗ v αg v βh , , ∂ p ξ βα = (0 , , Ψ αβ ⊗ v βg v αh , , ∂ p ξ ββ = (0 , , , Ψ αα ⊗ v βg v βh ) and exp ∗ f,p (Ψ ββ ) = exp ∗ f,p (Ψ βα ) = exp ∗ f,p (Ψ αβ ) = exp ∗ f,p (Ψ αα ) = 1 . Remark . Notice that the basis (2.26) depends on the choice of the L -basis v αg , v βg of V g and the L -basis v αh , v βh of V h . Then each element of the basis { ξ αα , ξ αβ , ξ βα , ξ ββ } depends on this choice upto multiplication by an element of L × .3. Special value formula for the triple product p -adic L -function in rank V := V f ⊗ V g ⊗ V h is the tensor product of the p -adic representations attached to the newforms f ∈ S ( N f ) Q , g ∈ M ( N g , χ ) L , h ∈ M ( N h , ¯ χ ) L , and we assume from now on that gcd( N f , N g , N h ) is square free. Recall that V ❛ g (resp. V ❛ h ) standsfor the Artin representation attached to g (resp. h ) and ρ denotes the tensor product representation ρ : Gal( H/ Q ) −→ GL( V ❛ g ⊗ V ❛ h ) ∼ = GL ( L ) . The complex L -function L ( E, ρ, s ) = L ( f ⊗ g ⊗ h, s )has entire continuation and satisfies a functional equation relating the value at s with the value at2 − s . Let ǫ be the sign of this functional equation and denote N := lcm( N f , N g , N h ). Then ǫ isthe product of local signs ǫ = Y v ǫ v , where v runs over the places of Q dividing N or ∞ . In this setting, ǫ ∞ = +1. Assume also that ǫ v = +1 for all v | N . In particular, the global sign is ǫ = 1 and the order of vanishing of L ( E, ρ, s )at the central point s = 1 is even. RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 12 Recall that p stands for a prime that does not divide N , and that g ∈ Λ g [[ q ]] (resp. h ∈ Λ h [[ q ]])is a Hida family passing through the p -stabilization g α (resp. h α ) such that U p g α = α g g α (resp. U p h α = α h h α ). Similarly, denote by f ∈ Λ f [[ q ]] a Hida family passing through the p -stabilization f α of f .Denote by L gp ( f , g , h ) the triple product p -adic L -function defined in [DR17], attached to thechoice of Λ-adic test vector (˘ f , ˘ g , ˘ h ) of [Hsi17, Chap. 3]. The values L gp ( f , g , h )( k, ℓ, m ) of this p -adic L -function at triples of integers ( k, ℓ, m ) with ℓ ≥ k + m interpolate the square root of thealgebraic part of L (˘ f k ⊗ ˘ g ℓ ⊗ ˘ h m , k + ℓ + m −
22 ) , (3.1)where ˘ f k , ˘ g ℓ , ˘ h m denote the specializations of ˘ f , ˘ g , ˘ h at weights k, ℓ, m .There is an analogous triple product p -adic L -function L fp ( f , g , h ) that interpolates (3.1) but forthe values ( k, ℓ, m ) with k ≥ ℓ + m . In particular, L fp ( f , g , h )(2 , ,
1) is directly related to L ( E, ρ, L gp ( f , g , h )(2 , ,
1) when L ( E, ρ, = 0. In particular, theElliptic Stark Conjecture predicts that when E ( H ) ρ is 2-dimensional L gp ( f , g , h )(2 , ,
1) is relatedto the p -adic logarithms of global elements in E ( H ) ρ .In the present note, our running Assumption 2.4 is that Sel p ( Q , V ) = 0. This implies that E ( H ) ρ = 0 and, conjecturally, it also implies that L ( E, ρ, = 0. The main result of this section isan explicit formula for L gp ( f , g , h )(2 , ,
1) in this case, and this can be seen as completing the studyof L gp ( f , g , h )(2 , ,
1) initiated in [DLR15]. The main tool that we will use are the generalized Katoclasses κ := κ ( f, g α , h α ) ∈ Sel ( p ) ( Q , V )introduced in [DR17, § κ and the p -adic L -values L fp ( f , g , h )(2 , ,
1) and L gp ( f , g , h )(2 , , L fp := L fp ( f , g , h )(2 , ,
1) and L gp := L gp ( f , g , h )(2 , , π αβ : H ( Q p , V ) −→ H ( Q p , V αβ )be the projection map induced by the natural map V → V αβ . Proposition 3.1 (Darmon–Rotger) . (1) The element ∂ p κ lies in the image of the natural map H s ( Q p , V ββ ) −→ H s ( Q p , V ) . In particular, ∂ p κ can be viewed as an element of H s ( Q p , V ββ ) . Moreover, (3.2) exp ∗ ββ ( ∂ p κ ) = 2(1 − pα f α − g α − h ) α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f α g α − h ) × L fp . (2) The element π αβ res p κ ∈ H ( Q p , V αβ ) belongs to H f ( Q p , V αβ ) , and (3.3) log αβ ( π αβ res p κ ) = L gp × − χ ( p ) p − α f a p ( g ) − a p ( h )) − . Proof.
The fact that ∂ p κ is the image of an element in H s ( Q p , V ββ ) is [DR17, Proposition 2.8].The equality (3.2) follows from Proposition 5.2 and Theorem 5.3 of [DR17]. By part (1) of theproposition π sαβ ∂ p κ = 0 in the singular quotient H s ( Q p , V αβ ). This means that π αβ res p κ belongsto H f ( Q p , V αβ ). Equality (3.3) follows from Proposition 5.1, Theorem 5.3 of [DR17]. (cid:3) RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 13 Using the κ recalled above and the basis (2.26) of Sel ( p ) ( V ), we can give a precise formula for L gp in the rank 0 setting. Define R βα ∈ E ( H p ) βα by the equality(3.4) π αβ res p ξ ββ = R βα ⊗ v αg v βh ∈ H f ( Q p , V αβ ) = ( E ( H p ) βα ⊗ V αβgh ) G Q p . Theorem 3.2.
The class κ is a multiple of ξ ββ ; more precisely, κ = Θ g Θ h − pα f α − g α − h ) L fp α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f a p ( g ) a p ( h ) − ) · ξ ββ . Moreover, if we define the quantities L g α := Ω g Θ g , E := (1 − χ ( p ) p − α − g α h )(1 − pα f α − g α − h ) α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f α g α − h ) then we have that L gp = E × log p ( R βα ) L g α × L fp mod L × . Proof.
By Proposition 3.1, κ is an element of Sel ( p ) ( Q , V ) such that(3.5) exp ∗ ( ∂ p κ ) = (0 , , , − pα f α − g α − h ) α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f α g α − h ) × L fp ) . Then κ is a multiple of the element ξ ββ ; indeed κ = exp ∗ ββ ( ∂ p κ )exp ∗ ββ ( ∂ p ξ ββ ) ξ ββ . Observe that (3.5) gives us the expression for the numerator. We now compute the denominator.exp ∗ ββ ( ∂ p ξ ββ ) = h exp ∗ f,p (Ψ αα ) ⊗ v βg v βh , ω g ω h i = exp ∗ f,p (Ψ αα )Θ g Θ h = 1Θ g Θ h . Here we used the fact that η g η h = Θ g Θ h v βg v βh . So we get κ = exp ∗ ββ ( ∂ p κ )exp ∗ ββ ( ∂ p ξ ββ ) · ξ ββ = 2(1 − pα f α − g α − h )Θ g Θ h α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f α g α − h ) × L fp · ξ ββ RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 14 By (3.3), L gp = 12 (1 − χ ( p ) p − α − g α h ) log αβ ( π αβ res p κ )= Θ g Θ h (1 − χ ( p ) p − α − g α h )(1 − pα f α − g α − h ) α g α h (1 − α − f α g α h )(1 − χ − ( p ) α − f α g α − h ) × L fp log αβ ( π αβ res p ξ ββ )= E Θ g Θ h L fp log αβ ( π αβ res p ξ ββ )= E Θ g Θ h L fp h log p ( R βα ) ⊗ v αg v βh , η g ω h i = EL fp Θ g Θ h Ω g Θ h log p ( R βα )= EL fp Θ g Ω g log p ( R βα )= EL fp L g α log p ( R βα )since ω g η h = Ω g Θ h ⊗ v αg v βh . (cid:3) We end this section by noting that L g α is often expected to be related to the Gross–Stark Unit u g α attached to the modular form g α as defined in [DLR15, § g is not the theta series of a Hecke character of a real quadratic field in which p splits, [DR16, Conjecture 2.1] predicts that L g α ? = log p ( u g α ) mod L × . (3.6)Thus we obtain the following consequence of Theorem 3.2, under the aforementioned hypothesis: Corollary 3.3.
Assuming the equality (3.6) , if
Sel p ( Q , V ) = 0 then L gp = E × log p ( R βα )log p ( u g α ) × L fp . The case of theta series of an imaginary quadratic field K where p splits In this section we will consider a particular case where g and h are theta series of the sameimaginary quadratic field in which p splits. We will see that in this setting the representation V decomposes in a way that forces L gp to vanish when the complex L -function does not vanish at thecentral critical point; that is, the special value of the p -adic L -function vanishes in analytic rank 0.Let K be an imaginary quadratic field of discriminant D K . Let ψ g , ψ h : A × K → C × be twofinite order Hecke characters of K of conductors c g , c h and central characters ε, ¯ ε respectively. Here ε : A × Q → C × is a finite order character of and ¯ ε denotes is complex conjugate. Let g and h be thetheta series attached to ψ g and ψ h . They are modular forms of weight one, and their levels andnebentype characters are given by N g := D K · N K ( c g ) , N h := D K · N K ( c h ) , χ := χ K · ε, ¯ χ = χ K · ¯ ε, where N K stands for the norm on ideals of K and we regard ε and ¯ ε as Dirichlet characters viaclass field theory. That is to say, g ∈ M ( N g , χ ) , and h ∈ M ( N h , ¯ χ ) . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 15 Let f ∈ S ( N f ) be a newform with rational coefficients and let E be the associated elliptic curveover Q . We will particularize some of the results of the previous sections to this choice of forms f , g , and h , so we will use the same notations as before. In particular, ρ stands for the Artinrepresentation afforded by V g ⊗ V h and p is a prime that does not divide N f · N g · N h . In thissection, we will make the following additional assumptions:(1) gcd( N f , c g c h ) = 1;(2) p splits in K .A finite order Hecke character ψ of K can be regarded, via class field theory, as a Galois character ψ : G K → A × K . Let σ be any element in G Q \ G K . We denote by ψ ′ the character defined by ψ ′ ( σ ) := ψ ( σ σσ − ) (this does not depend on the particular choice of σ ). Also, ψ gives rise to a1-dimensional representation of G K , and we let V ψ = Ind Q K ( ψ ) denote the induced representation; itis a 2-dimensional representation of G Q . Observe that, with this notation, V g = V ψ g and V h = V ψ h .There is a well-known decomposition of V g ⊗ V h as the direct sum of two representations:(4.1) V g ⊗ V h = V ψ ⊕ V ψ , where the characters ψ and ψ are ψ := ψ g ψ h , and ψ := ψ g ψ ′ h . This induces a decomposition of the representation V = V f ⊗ V g ⊗ V h as a direct sum of tworepresentations:(4.2) V = V ⊕ V , where V := V f ⊗ V ψ , and V := V f ⊗ V ψ . This induces a factorization of complex L -functions L ( E, ρ, s ) = L ( E, ψ , s ) · L ( E, ψ , s ) . Under our assumption that gcd( N f , c g c h ) = 1 the local signs of L ( E, ψ , s ) and L ( E, ψ , s ) areequal, so that the local signs of L ( E, ρ, s ) are all equal to +1 and therefore the assumption on localsigns of Section 3 is satisfied.
Theorem 4.1.
In the setting of this section, if L ( E, ρ, = 0 then L gp = 0 . Proof. If L ( E, ρ, = 0 then L ( E, ψ i , = 0 for i = 1 ,
2. Note that ψ and ψ are ring classcharacters of the imaginary quadratic field K . Then, by results of Gross–Zagier and Kolyvagin(4.3) Sel p ( Q , V i ) = 0 for i = 1 , . The decomposition (4.2) induces a decomposition of the Selmer groups(4.4) Sel p ( Q , V ) = Sel p ( Q , V ) ⊕ Sel p ( Q , V ) , and analogously for the relaxed and the strict Selmer groups of V . In particular, Sel p ( Q , V ) = 0 . Since p splits in K we can write p O K = p ¯ p , and from our assumption that p ∤ N f · N g · N h wesee that p ∤ c g c h . Without loss of generality we can suppose that ψ g ( p ) = α g , ψ g (¯ p ) = β g , ψ h ( p ) = α h , ψ h (¯ p ) = β h , so that V = V αα ⊕ V ββ and V = V αβ ⊕ V βα . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 16 By (4.3), the same computations as in § ( p ) ( Q , V ) ∂ p −→ H s ( Q p , V ) ( π sαα ,π sββ ) −→ H s ( Q p , V αα ) ⊕ H s ( Q p , V ββ ) , where π sαα denotes the natural map in the singular quotient induced by the projection V → V αα ,and analogously for π sββ . Similarly, there are dual exponential mapsexp ∗ αα : H s ( Q p , V αα ) = H ( Q p , V − f ⊗ V ααgh ) −→ L p and exp ∗ ββ : H s ( Q , V ββ ) = H ( Q p , V − f ⊗ V ββgh ) −→ L p which are in fact isomorphisms.Then Sel ( p ) ( Q , V ) has dimension 2 over Q p with the canonical basis ζ αα , ζ ββ , where ζ αα is characterized (up to scalars in L × ) by the fact thatexp ∗ αα ( π αα ∂ p ( ζ αα )) = 1 , and exp ∗ ββ ( π ββ ∂ p ( ζ αα )) = 0 . Similarly, exp ∗ αα ( π αα ∂ p ( ζ ββ )) = 0 , and exp ∗ ββ ( π ββ ∂ p ( ζ ββ )) = 1 . Analogously, Sel ( p ) ( Q , V ) has dimension 2 with basis ζ αβ , ζ βα .By Theorem 3.2, the value L gp is a multiple of log αβ (res p ξ ββ ). On the other hand, using thedecomposition Sel ( p ) ( Q , V ) = Sel ( p ) ( Q , V ) ⊕ Sel ( p ) ( Q , V ) , the element ξ ββ ∈ Sel ( p ) ( Q , V ) corresponds to a multiple of (0 , ζ ββ ), and this implies that π βα res p ξ ββ = 0 . (cid:3) Numerical computations
In this section we present some numerical examples. They have been computed with a Sage([S + github.com/mmasdeu/ellipticstarkconjecture . The data for theweight-one modular forms can be found in Alan Lauder’s website. Dihedral case. ( a ) We computed L gp ( f , g , g )(2 , ,
1) with f the Hida family passing through the modular form f E of weight 2 attached to an elliptic curve E/ Q of conductor N f and g attached to the weight-onemodular form g = θ (1 K ) for some imaginary quadratic field K . The modular form g belongsthen to M ( N g , χ K ) Q . For each of the entries in the table we give the Cremona label forthe elliptic curve E f , its conductor N f , the field K , the level N g of g , the level N such that p α N = lcm( N f , N g ) with α ≥ p ∤ N . In all of these cases, we obtained L gp ( f , g , g ) = 0up to the working precision of p . Due to computational restrictions, only in the ramified casewe have been able to compute examples where p divides the conductor of the elliptic curve.Note that all the elliptic curves arising in Table 1 below have rank 0 over K , and thus thezeros obtained in this table are accounted for by Theorem 4.1. See http://people.maths.ox.ac.uk/lauder/weight1/ . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 17 E f K N g p N L gp ( f , g , g )11a Q ( √−
5) 20 7 220 011a Q ( √−
11) 11 5 11 019a Q ( √−
19) 19 5 19 019a Q ( √−
19) 19 7 19 039a Q ( √−
39) 39 5 39 051a Q ( √−
51) 51 5 51 055a Q ( √−
55) 55 7 55 0187a Q ( √− Table 1.
Cases with p split in K .In the next two tables we see instances of zeros which we expect are explained by the signof the action of the level N Atkin-Lehner operator although we have not verified this in detail. E f K N g p N L gp ( f , g , g )11a Q ( √−
3) 3 5 33 011a Q ( √−
11) 11 7 11 015a Q ( √−
15) 15 7 15 039a Q ( √−
39) 39 7 39 051a Q ( √−
51) 51 7 51 067a Q ( √−
67) 67 5 67 067a Q ( √−
67) 67 7 67 0187a Q ( √− Table 2.
Cases with p inert in K . E f K N g p N L gp ( f , g , g )15a Q ( √−
15) 15 5 3 035a Q ( √−
35) 35 5 7 035a Q ( √−
35) 35 7 5 055a Q ( √−
55) 55 5 11 0
Table 3.
Cases with p ramified in K .In what follows we illustrate with examples the fact that the quantity L gp ( f , g , g ) is notalways zero.( b ) In this example we fix f to be attached to the elliptic curve E f : y = x + x − x + 18, ofconductor N f = 120. The weight-one form g we consider has level N g = 120 also, and has RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 18 q -expansion g ( q ) = q + iq + iq − q − iq − q − iq − q + q − iq + q + q − iq + iq + q − q − iq + iq − q + iq + O ( q ) , where i = −
1. It is the theta series attached to the Dirichlet character ǫ modulo 120 definedby ǫ (97) = − , ǫ (31) = 1 , ǫ (41) = − , ǫ (61) = − . The field cut out by ǫ is K = Q ( √− p = 5 which is split in both L = Q ( √− K . Note that p divides N f and N g . We compute to precision 10 the quantity L g ( f , g , g )(2 , ,
1) = 4 · · + 4 · + 3 · + 4 · + 3 · + 5 + 2 · + O (5 ) . With the same setting, we take p = 13 (now p is split in L but inert in K ). We obtain L g ( f , g , g )(2 , ,
1) = 7 + 3 ·
13 + 10 · + 13 + 11 · + 13 + 6 · + 4 · + 5 · + O (13 )( c ) Let E f be the elliptic curve y + y = x + x + 42 x −
131 with label . It has conductor N f = 175 and rank 0. Let g = h be the theta series of the character ǫ of K = Q ( α ) with α satisfying α − α + 2 = 0, of discriminant D K = − O K (which is inert, ofnorm 25), satisfying ǫ (127) = − , ǫ (101) = − . The modular form g has q -expansion g ( q ) = q + iq − iq + iq − q − q + q − q − iq − iq − iq + q + O ( q ) , where again i = 1. For p = 13 (which is inert in K and split in L ), we obtain L g ( f , g , g )(2 , ,
1) = 1 + 3 ·
13 + 2 · + 13 + 12 · + 9 · + 3 · + 5 · + O (13 ) . ( d ) Finally, consider the elliptic curve E f of conductor 175 from the previous example, and for g = h consider the theta series of another character ǫ of K = Q ( α ), α − α + 2 = 0, ofdiscriminant D K = − O K (inert, of norm 25), now taking the values ǫ (127) = 1 , ǫ (101) = − . This yields a modular form g with q -expansion g ( q ) = q + q − q − q + q − q − q − q + q − q + q − q + O ( q ) . We numerically obtain for p = 13 that L g ( f , g , g )(2 , ,
1) = 0 . Again, we do not have a way to prove that L g ( f , g , g )(2 , ,
1) is actually zero.5.2.
Exotic image case.
In the non-CM setting, we have been able to compute the followingexample. Consider E f : y = x − x −
27, which has conductor N f = 124. Let g be the modularform of level N g = 124 and projective image A , defined as the theta series of the character ǫ ofconductor 124 having values ǫ (65) = α − , ǫ (63) = − , where α satisfies α − α + 1 = 0. The modular form g has q -expansion g ( q ) = q − α q + (cid:0) − α + α (cid:1) q − q + (cid:0) α − (cid:1) q − α q + (cid:0) α − α (cid:1) q + α q + αq − αq + (cid:0) α − α (cid:1) q + (cid:0) − α + 1 (cid:1) q + α q + α q + q − α q + (cid:0) − α + α (cid:1) q + (cid:0) − α + 1 (cid:1) q + (cid:0) α − (cid:1) q + (cid:0) α − (cid:1) q + α q − αq + α q + O ( q ) . RIPLE PRODUCT p -ADIC L -FUNCTIONS AND NON-CRYSTALLINE CLASSES 19 We let h = g ∗ its complex conjugate, and compute with p = 13, obtaining L g ( f , g , h )(2 , ,
1) = 1+5 · · +4 · +6 · +6 · +6 · +13 +3 · +9 · +9 · + O (13 ) . References [Bel] Jo¨el Bela¨ıche. An introduction to the conjecture of Bloch and Kato.
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