p -adic Asai L -functions attached to Bianchi cusp forms
aa r X i v : . [ m a t h . N T ] A ug p -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSPFORMS BASKAR BALASUBRAMANYAM, EKNATH GHATE, AND RAVITHEJA VANGALA
Abstract.
We establish a rationality result for the twisted Asai L -values attached toa Bianchi cusp form and construct distributions interpolating these L -values. Using themethod of abstract Kummer congruences, we then outline the main steps needed toshow that these distributions come from a measure. Preliminaries
Let F be an imaginary quadratic field with ring of integers O F . Write F = Q ( √− D )with D > − D the discriminant of F . Let S k ( n ) denote the space of Bianchi cuspforms of weight k = ( k, k ), k ≥
2, and level n and central character with trivial finite partand infinity type (2 − k, − k ). Let f ∈ S k ( n ) be a normalized eigenform and let c ( m , f ) bethe Fourier coefficients of f , for any integral ideal m ⊂ O F . The eigenform f correspondsto a tuple ( f , . . . , f h ) of classical Bianchi cusp forms, where h is the class number of F .We take f = f ∈ S k (Γ ( n )) and only focus on this since the Asai L -function dependsonly on f .Let c ( r ), for r ≥
1, denote c (( r ) , f ). Define the Asai L -function of f by the formula G ( s, f ) = L N (2 s − k + 2 , ∞ X r =1 c ( r ) r s , where N is the positive generator of the ideal n ∩ Z and L N ( s,
11) is the L -functionattached to the trivial character modulo N . The special values of this function areinvestigated in [Gha99]. A generalization to cusp forms defined over CM fields can befound in [Gh99b].Let p ∈ Z be an odd prime integer that is relatively prime to N and that is alsounramified in F . Let χ : ( Z /p j Z ) × → C × be a Dirichlet character with conductordividing p j . Define the twisted Asai L -function of f by the formula G ( s, χ, f ) = L N (2 s − k + 2 , χ ) ∞ X r =1 c ( r ) χ ( r ) r s . This has an Euler product expansion G ( s, χ, f ) = Y p G p ( s, χ, f ) , where the local L -functions at all but finitely many primes are described as follows. Let l = p be an integer prime, not dividing N . For any l | l , let α ( l ) and α ( l ) denote thereciprocal roots of the Hecke polynomial of f at l : 1 − c ( l , f ) X + Nm( l ) k − X . Then1 G l ( s, χ, f ) = Q i,j (1 − χ ( l ) α i ( l ) α j (¯ l ) l − s ) if l = l ¯ l , (1 − χ ( l ) α ( l ) l − s )(1 − χ ( l ) l − s +2 k − )(1 − χ ( l ) α ( l ) l − s ) if l = l is inert , (1 − χ ( l ) α ( l ) l − s )(1 − χ ( l ) l − s + k − )(1 − χ ( l ) α ( l ) l − s ) if l = l is ramified . We want to find ‘periods’ and prove that the special values of the twisted Asai L -functions are algebraic after dividing by these periods. We also want to p -adically inter-polate the special values of G ( s, χ, f ) as χ varies over characters of p -power conductor.2. Complex valued distributions
Following Panchishkin, we now construct a complex valued distribution that is relatedto the twisted Asai L -function. This section basically follows Coates–Perrin-Riou [CP89]and Courtieu–Panchishkin [CP04, § G ( s, f ) has an Euler product formula G ( s, f ) = Y p G p ( s, f ) = ∞ X r =1 d ( r ) r s , and hence satisfies the hypothesis in the above references. We now assume that our fixedprime p splits as p ¯ p in F . A similar argument will also work for p inert. Then the localEuler factor at p is of the form G p ( s, f ) = F ( p − s ) − where F ( X ) = (1 − α ( p ) α (¯ p ) X )(1 − α ( p ) α (¯ p ) X )(1 − α ( p ) α (¯ p ) X )(1 − α ( p ) α (¯ p ) X ) . In what follows we shall assume that f is totally ordinary at p . Hence we may assume,by possibly switching the subscripts i = 1, 2, that the inverse root κ := α ( p ) α (¯ p ) of thepolynomial F ( X ) is a p -adic unit. Also define a polynomial H ( X ) as H ( X ) = (1 − α ( p ) α (¯ p ) X )(1 − α ( p ) α (¯ p ) X )(1 − α ( p ) α (¯ p ) X ) . Let B = 1 and define B , B and B such that H ( X ) = 1 + B X + B X + B X . Let χ : ( Z /p j Z ) × → C × be a character with conductor C χ = p j χ . We want to define acomplex valued distribution that interpolates the values of the twisted L -function: G ( s, χ, f ) = ∞ X r =1 d ( r ) χ ( r ) r s . Define functions P s : Q → C by the formula P s ( b ) = ∞ X r =1 d ( r ) e πirb r s which converges absolutely for ℜ ( s ) sufficiently large. Define a distribution ˜ µ on Z × p bythe formula ˜ µ s ( a + p j Z p ) = p j ( s − κ j X i =0 B i P s ( ap i /p j ) p − is . We need to check that this satisfies the distribution relations. We will do this by showingthat(1) X a mod p j χ ( a )˜ µ s ( a + p j Z p )is independent of j as long as j ≥ j χ . For any character χ and integer M , define thegeneralized Gauss sum G M,p j = X a mod p j χ ( a ) e πiaM/p j . -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 3 It can be verified that G M,p j = (cid:26) p j − j χ G ( χ ) ¯ χ ( M/p j − j χ ) if p j − j χ | M , where G ( χ ) = G ,p jχ is the Gauss sum of χ .From the definition, we can write the quantity in equation (1) as p j ( s − κ j X a mod p j χ ( a ) X i =0 B i p − is ∞ X r =1 d ( r ) e πiap i r/p j r − s = p j ( s − κ j X i,r B i p − is d ( r ) r − s G p i r,p j = p js − j χ κ j G ( χ ) X i,r B i p − is d ( r ) r − s ¯ χ (cid:18) p i rp j − j χ (cid:19) Z (cid:18) p i rp j − j χ (cid:19) . (2)Here 11 Z is the characteristic function of integers. It appears since the only terms thatcontribute to the sum are those with p j − j χ | p i r (this follows from the above formula for G p i r,p j ). Now write each r = r r , where r is the p -power part and r is the away from p -part of r respectively. We know that¯ χ (cid:18) p i rp j − j χ (cid:19) = ¯ χ ( r ) ¯ χ (cid:18) p i r p j − j χ (cid:19) , and11 Z (cid:18) p i rp j − j χ (cid:19) = 11 Z (cid:18) p i r p j − j χ (cid:19) . Using these in equation (2), we get the expression(3) p js − j χ κ j G ( χ ) X r ¯ χ ( r ) d ( r ) r − s ! X i,r B i p − is d ( r ) r − s ¯ χ (cid:18) p i r p j − j χ (cid:19) Z (cid:18) p i r p j − j χ (cid:19) . Note that here we have also used the fact that d ( r ) = d ( r ) d ( r ).We also know that X r d ( r ) r − s = F ( p − s ) − , where the sum is taking over all powers of p . Moreover, we also have( X i B i p − is ) F ( p − s ) − = H ( p − s ) F ( p − s ) − = (1 − κp − s ) − . Hence X i,r B i p − is d ( r ) r − s = X r κ ord p r r − s , where the r varies over all powers of p . We also have the relation κ ord p r = X r = p i r B i d ( r ) . Hence setting r = p i r , we see that the only terms that contribute to the sum in equation(3) are those p -powers r of the form p j − j χ r for some p -power r . Also note that as r varies over all positive integers prime to p , we get X r ¯ χ ( r ) d ( r ) r − s = G ( s, ¯ χ, f ) . B. BALASUBRAMANYAM, E. GHATE, AND RAVITHEJA VANGALA
We remark that if χ is the trivial character of Z × p , then G ( s, ¯ χ, f ) is just the p -deprivedAsai L -function G p ( s, f ) − G ( s, f ), where G p ( s, f ) is the local Euler factor at p , since thefunction on Z induced by the trivial character χ is taken to vanish on p Z . In any case,we can rewrite equation (3) as p js − j χ κ j G ( χ ) G ( s, ¯ χ, f ) X r κ ord p r r − s ¯ χ (cid:18) r p j − j χ (cid:19) Z (cid:18) r p j − j χ (cid:19) = p js − j χ κ j G ( χ ) G ( s, ¯ χ, f ) X r κ j − j χ κ ord p r p − s ( j − j χ ) r − s ¯ χ ( r )= p j χ ( s − κ j χ G ( χ ) G ( s, ¯ χ, f ) X r κ ord p r r − s ¯ χ ( r )= p j χ ( s − κ j χ G ( χ ) G ( s, ¯ χ, f ) , since ¯ χ ( r ) = 0, unless r = 1, since by convention all Dirichlet characters of Z × p , includingthe trivial character, are thought of as functions on Z by requiring that they vanish on p Z . This simultaneously checks the distribution relations and establishes the relationship(4) Z χ d ˜ µ s = p j χ ( s − κ j χ G ( χ ) G ( s, ¯ χ, f )between these measures and twisted Asai L -values.We remark here that these calculations hold only for s ∈ C where G ( s, f ) is absolutelyconvergent. And this is known for all s such that ℜ ( s ) > k + 1, in view of the Heckebound c ( l , f ) = O (Nm( l ) k/ ), for all but finitely many primes l of F .In order to construct a measure, we need to show that this is a bounded distribution(after possibly dividing by some periods). We now modify the distribution ˜ µ s to constructthe distribution µ s ( a + p j Z p ) = ˜ µ s ( a + p j Z p ) + ˜ µ s ( − a + p j Z p ) . The distribution relations for µ s follows from those of ˜ µ s . Moreover Z χ dµ s = (cid:26) R χ d ˜ µ s if χ is even,0 if χ is odd . In the next section, we shall prove that the values of the distribution R χ dµ s in(4), for specific values of s , are rational, after dividing by some periods. In Section 6,we conjecture that these values are even p -adically bounded. We then conjecture thatthese values satisfy the so called abstract Kummer congruences, and hence come from ameasure. 3. Rationality result for twisted Asai L -values Let n = k − n = ( n, n ). For any O F -algebra A , let L ( n, A ) denote the setof polynomials in 4 variables ( X, Y, X, Y ) with coefficients in A , which are homogeneousof degree n in ( X, Y ) and homogeneous of degree n in ( X, Y ). We define an action ofSL ( A ) on this set by γ · P ( X, Y, X, Y ) = P ( γ ι ( X, Y ) t , ¯ γ ι ( X, Y ) t )= P ( dX − bY, − cX + aY, ¯ d X − ¯ b Y , − ¯ cX + ¯ aY ) , -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 5 for γ = (cid:20) a bc d (cid:21) ∈ SL ( A ) and where γ ι = det( γ ) γ − is the adjoint matrix of γ . Let Γ ( n )and Γ ( n ) be the usual congruence subgroups of SL ( O F ) with respect to the ideal n . Let H = { ( z, t ) | z ∈ C and t ∈ R with t > } be the hyperbolic upper half-space in R . There is an action of SL ( O F ) on H whichis induced by identifying H with SL ( C ) / SU ( C ) = GL ( C ) / [SU ( C ) · C × ]. The lastidentification is given by a transitive action of SL ( C ) on H defined via g g · ǫ , for g ∈ SL ( C ) and ǫ = (0 , ∈ H . We will let L ( n, A ) denote the system of local coefficientsassociated to L ( n, A ). So L ( n, A ) is the sheaf of locally constant sections of the projectionΓ ( n ) \ ( H × L ( n, A )) −→ Γ ( n ) \H . Analogous to the Eichler-Shimura isomorphism for classical cusp forms, there are isomor-phisms δ q : S k (Γ ( n )) −→ H qcusp (Γ ( n ) \H , L ( n, C )) , for q = 1 ,
2. Here the cohomology on the right is cuspidal cohomology with local coeffi-cients. We take δ = δ since we are interested in 1-forms.Let us now describe the image of δ ( f ) under this map. Let γ ∈ SL ( C ) and z = ( z, t ) ∈H . After identifying z with the matrix (cid:20) z − tt ¯ z (cid:21) , recall that the action of γ on z is by γ · z = [ ρ ( a ) z + ρ ( b )][ ρ ( c ) z + ρ ( d )] − , where ρ ( α ) = (cid:20) α
00 ¯ α (cid:21) . Define the automorphy factor j ( γ, z ) = ρ ( c ) z + ρ ( d ) ∈ GL ( C ).Let L (2 n + 2 , C ) denote the space of homogeneous polynomials of degree 2 n + 2 in twovariables ( S, T ) and coefficients in C . We will consider L (2 n + 2 , C ) with a left action ofSL ( C ).Recall that f is a function H → L (2 n + 2 , C ) that satisfies the transformation property f ( γz, ( S, T )) = f ( z, t j ( γ, z )( S, T ) t ) , for γ ∈ Γ ( n ). There is a related ‘cusp form’ F : SL ( C ) → L (2 n + 2 , C ) on SL ( C ) whichis defined by the formula f ( z, ( S, T ) t ) = F ( g, t j ( g, ǫ )( S, T ) t ) , where g ∈ SL ( C ) is chosen such that g · ǫ = z .By Clebsch-Gordon, there is an SU ( C )-equivariant homomorphismΦ : L (2 n + 2 , C ) ֒ → L ( n, C ) ⊗ L (2 , C ) . Then δ ( f ) can explicitly be described as [Gha99, (13)](5) δ ( f )( g ) = g · (Φ ◦ F ( g )) , ∀ g ∈ SL ( C ) . Note that here the action of g on L ( n, C ) is as described above. But the action of SL ( C )on L (2 , C ) is identified with the natural action of SL ( C ) on Ω ( H ) = C dz ⊕ C dt ⊕ C d ¯ z (see [Gha99, (6)]). The identification of L (2 , C ) with Ω ( H ) is given by sending A dz, AB
7→ − dt and B
7→ − d ¯ z . With this identification, we view δ ( f ) as a L ( n, C )valued differential form. It is also invariant under the action of SU ( C ) [Gha99, (14)], soit descends to a vector valued 1-form on H . Moreover, if γ ∈ Γ ( n ), then δ ( f )( γz ) = γ · ( δ ( f )( z )) . So δ ( f ) descends to an element of H cusp (Γ ( n ) \H , L ( n, C )). B. BALASUBRAMANYAM, E. GHATE, AND RAVITHEJA VANGALA
We make this formula more explicit. Let U and V be auxiliary variables and definethe following homogeneous polynomial of degree 2 n + 2 by Q = (cid:18)(cid:18) n + 2 α (cid:19) ( − n +2 − α U α V n +2 − α (cid:19) α =0 ,..., n +2 . Define ψ ( X, Y, X, Y , A, B ) = [ ψ ( X, Y, X, Y , A, B ) , . . . , ψ n +2 ( X, Y, X, Y , A, B )] t by theformula ( XV − Y U ) n ( XU + Y V ) n ( AV − BU ) = Q · ψ, where each ψ i is a polynomial that is homogeneous of degree n in ( X, Y ), homogeneousof degree n in ( X, Y ) and homogeneous of degree 2 in (
A, B ).Let z ∈ H , then since SL ( C ) acts transitively on H , there is a g ∈ SL ( C ) such that z = g · ǫ where ǫ = (0 , ∈ H . Let F α : SL ( C ) → C , for α = 0 , . . . , n + 2, be thecomponents of the function F : SL ( C ) → L (2 n + 2 , C ). Then(Φ ◦ F )( g ) = [ F ( g ) , . . . , F n +2 ( g )] · ψ ( X, Y, X, Y , A, B )and δ ( f )( z ) = g · (Φ ◦ F ( g )) = [ F ( g ) , . . . , F n +2 ( g )] · ψ ( g ι ( X, Y ) t , ¯ g ι ( X, Y ) t , t j ( g − , ǫ ) − ( A, B ) t ) , where A , AB, B are replaced by dz, − dt, − d ¯ z .Now, let β ∈ F ⊂ C and let T β denote the translation map H → H given by sending z = ( z, t ) ( z + β, t ). When we view H as a quotient space of SL ( C ), this map isinduced by sending the coset g SU ( C ) γ β g SU ( C ), where γ β := (cid:20) β (cid:21) . Let Γ β ( n ) := γ − β Γ ( n ) γ β ⊂ SL ( F ). Notice that if Γ β ( n ) g ′ SU ( C ) = Γ β ( n ) g SU ( C ), thenΓ ( n ) γ β g ′ SU ( C ) = Γ ( n ) γ β g SU ( C ). Thus the translation map T β induces a well-definedmap Γ β ( n ) \H T β −→ Γ ( n ) \H , which we again denote by T β .We now recall some basic facts about functoriality of cohomology with local coefficients.For i = 1 ,
2, let X i be topological spaces with universal covers ˜ X i and fundamental groupsΓ i (after fixing some base points). Let M i be local coefficient systems on X i , i.e., each M i is an abelian group with an action of the fundamental group Γ i . Let φ : X → X bea map between the spaces, it induces a map φ ∗ : Γ → Γ on the fundamental groups. Amap between the coefficient systems ˜ φ : M → M is said to be compatible with φ if itsatisfies γ ˜ φ ( m ) = ˜ φ ( φ ∗ ( γ ) m ) , ∀ m ∈ M and γ ∈ Γ . In other words, ˜ φ must be a map between representations when M is viewed as a repre-sentation of Γ via the map φ ∗ . For any compatible pair ( φ, ˜ φ ), there exists an inducedmap φ ∗ : H q ( X , M ) → H q ( X , M )at the level of cohomology. This map is constructed as follows. Let S ∗ ( ˜ X i ) denote thesingular complex of the universal covers. There is a natural action on the right by Γ i viadeck transformations. Given a singular q -simplex σ : ∆ q → ˜ X i and g ∈ Γ i , we convertthis right action into a left action by setting g · σ = σ · g − . The cohomology groups withlocal coefficients are given by the homology of the complex Hom Z Γ i ( S ∗ ( ˜ X i ) , M i ). Themap φ ∗ is induced by the following map on the complexes (which we again denote by φ ∗ ) φ ∗ : Hom Z Γ ( S q ( ˜ X ) , M ) → Hom Z Γ ( S q ( ˜ X ) , M ) . -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 7 Given a cochain C ∈ Hom Z Γ ( S q ( ˜ X ) , M ) and τ : ∆ q → ˜ X ∈ S q ( ˜ X ), define(6) φ ∗ ( C )( τ ) = ˜ φ ( C ( φ ◦ τ )) , where we continue to denote by φ : ˜ X → ˜ X the unique lift of φ : X → X tothe universal covers. This construction is independent of the base points chosen in thebeginning.Let us now apply this to our situation with X = Γ β ( n ) \H and X = Γ ( n ) \H . Inthis case, Γ = Γ β ( n ) and Γ = Γ ( n ) and they act on M = M = L ( n, C ) sincethey are subgroups of SL ( F ). We take the map φ to be the translation map T β whichinduces, at the level of fundamental groups, the map ( T β ) ∗ : Γ β ( n ) → Γ ( n ) which sends γ − β γγ β γ . It is an easy check that the map ˜ T β : M → M sending P γ − β P , for P ∈ L ( n, C ), is compatible with T β . By the discussion above this induces a map at thelevel of cohomology T ∗ β : H q (Γ ( n ) \H , L ( n, C )) → H q (Γ β ( n ) \H , L ( n, C )) . When q = 1, what is the image of the element δ ( f ) ∈ H (Γ ( n ) \H , L ( n, C ))? Aftertranslating the above map T ∗ β in terms of vector valued differential forms, we get that T ∗ β ( δ ( f ))( z ) (6) = γ − β δ ( f )( γ β z ) (5) = γ − β γ β g · (Φ ◦ F ( γ β g )) = g · (Φ ◦ F ( γ β g ))= [ F ( γ β g ) , . . . , F n +2 ( γ β g )] · ψ ( g ι ( X, Y ) t , ¯ g ι ( X, Y ) t , t j ( g − , ǫ ) − ( A, B ) t ) . Here z ∈ H and we take g ∈ SL ( C ) such that gz = ǫ , and A , AB, B are to be replacedby dz, − dt, − d ¯ z .Following [Gha99, § T ∗ β ( δ ( f )) | H where H = { x + it | x, t ∈ R and t > } is the usual upper half-plane which is embedded intothe hyperbolic 3-space H as x + it (cid:20) x − tt x (cid:21) . As in loc. cit. , we make the following two simplifications. Firstly, since we wish to computethis differential form on H , we set dz = d ¯ z in our computations. Secondly, we only needto calculate the differential form (cid:20) − x (cid:21) · (cid:0) T ∗ β ( δ ( f )) | H (cid:1) , so we set x = 0 in ψ and onlycalculate the modified differential form which we denote by ^ T ∗ β ( δ ( f )) | H . Note that thecomponents ψ α of ψ , for α = 0 , . . . , n + 2, are given by ψ α ( X, Y, X, Y , A, B ) = ( − α A c α − ABc α − + B c α − (cid:0) n +2 α (cid:1) , where c α ( X, Y, X, Y ) = n X j,k =0 n = α + j − k ( − k (cid:18) nj (cid:19)(cid:18) nk (cid:19) X n − k Y k X n − j Y j . For x, t ∈ R and t >
0, let g = √ t (cid:20) t x (cid:21) . Then g · ǫ = ( x, t ) ∈ H ⊂ H and g ι = ¯ g ι = √ t (cid:20) − x t (cid:21) . Moreover, j ( g − , ǫ ) − = j ( g, ǫ ) = √ t (cid:20) (cid:21) . Let f α , for α = 0 , . . . , n + 2,be the components of f : H → L (2 n + 2 , C ). The precise relationship between f α and F α is given by f α ( z ) = 1 √ t n +2 F α ( g ) . B. BALASUBRAMANYAM, E. GHATE, AND RAVITHEJA VANGALA
Note that if z = ( z, t ), then T β ( z ) = ( z + β, t ) does not affect the t coordinate. Hence f α ( T β z ) = 1 √ t n +2 F α ( γ β g ) . Using this and the pullback formula, we get ^ T ∗ β ( δ ( f )) | H = n +2 X α √ t n +2 f α ( T β z ) ψ α (cid:18) √ t X, √ t Y, √ t X, √ t Y , √ t A, √ t B (cid:19) = n +2 X α f α ( T β z ) ψ α ( X, tY, X, tY , A, B ) , where we replace ( A , AB, B ) by ( dx, − dt, − dx ). We have now constructed an element ^ T ∗ β ( δ ( f )) | H ∈ H cusp (Γ β ( N ) \ H , L ( n, C )) where Γ β ( N ) := Γ β ( n ) ∩ Γ ( N ) = Γ β ( n ) ∩ SL ( Z ),since in the latter matrix group, the lower left entries are divisible by N . As in [Gha99,see below Lemma 2], we have a decomposition of this cohomology group as(7) H cusp (Γ β ( N ) \ H , L ( n, C )) ∼ −→ n M m =0 H cusp (Γ β ( N ) \ H , L (2 n − m, C )) . We will call the projection of ^ T ∗ β ( δ ( f )) | H into the m -th component by ^ T ∗ β ( δ n − m ( f )),slightly abusing notation since the subscript 2 n − m should technically be outside theparentheses. For each m , define(8) g α ( z ) = f α ( z )+( − n +1 − α + m f n +2 − α ( z ) ( n +2 α ) if α = 0 , , . . . , n, f n +1 ( z ) ( n +2 n +1 ) if α = n + 1 . Then, we have(9) ^ T ∗ β ( δ n − m ( f ))( x, t ) = n − m X l =0 ( A l dx + 2 B l dt ) t n − m − l X l Y n − m − l , where A l = n +1 X α =0 ( − α g α ( T β ( x, t )) a ( m, l, α ) ,B l = n +1 X α =0 ( − α g α ( T β ( x, t )) b ( m, l, α ) , with a ( m, l, α ) and b ( m, l, α ) the integers defined at the end of [Gha99, § n ≥ Z [1 /n !]-algebra A , there is an SL ( Z )-equivariant pairing[Gha99, Lemma 4] h , i : L ( n, A ) ⊗ L ( n, A ) → A, which induces by Poincare duality a pairing h , i : H c (Γ β ( N ) \ H , L ( n, A )) ⊗ H (Γ β ( N ) \ H , L ( n, A )) → H c (Γ β ( N ) \ H , A ) → A, where the last map H c (Γ β ( N ) \ H , A ) → A is given by integrating a compactly supported2-form on a fundamental domain [Γ β ( N ) \ H ] of Γ β ( N ) \ H . We will use this pairing when A = C , A = E is a p -adic number field with p > n , and with A = O E , its ring of integers.When A = C , the pairing can be extended to h , i : H cusp (Γ β ( N ) \ H , L ( n, C )) ⊗ H (Γ β ( N ) \ H , L ( n, C )) → H cusp (Γ β ( N ) \ H , C ) → C ∪{∞} . -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 9 For each m , there is an Eisenstein differential form E β n − m +2 for Γ β ( N ) given by(10) E β n − m +2 ( s, z ) = X γ ∈ Γ β ( N ) ∞ \ Γ β ( N ) γ − · γ ∗ ( ω y s ) , where ω = ( X − zY ) n − m dz . One may check that E β n − m +2 ( s, z ) = X γ = ( a bc d ) ∈ Γ β ( N ) ∞ \ Γ β ( N ) cz + d ) n − m +2 | cz + d | s · y s ω. (11)We view E β n − m +2 as an element of H (Γ β ( N ) \ H , L ( n, C )). We now wish to evaluate h T ∗ β ( δ n − m ( f )) , E β n − m +2 i , following [Gha99, § h T ∗ β ( δ n − m ( f )) , E β n − m +2 i = Z [Γ β ( N ) \ H ] h T ∗ β ( δ n − m ( f ))( x, t ) , E β n − m +2 ( x, t ) i = Z [Γ β ( N ) \ H ] h ^ T ∗ β ( δ n − m ( f )) , ^ E β n − m +2 i , where the e indicates that we have twisted the differential forms by the action of thematrix (cid:20) − x (cid:21) . Using a standard unwinding argument, the last integral becomes Z ∞ Z h ^ T ∗ β ( δ n − m ( f )) , e ωt s i , where e ω = ( X − itY ) n − m dz . Using the expression (9) for ^ T ∗ β ( δ n − m ( f ))( x, t ) and thedefinition of the pairing, we have Z ∞ Z h ^ T ∗ β ( δ n − m ( f )) , e ωt s i = Z ∞ Z
10 2 n − m X l =0 i l +1 A l t n − m + s dxdt − Z ∞ Z
10 2 n − m X l =0 i l B l t n − m + s dxdt. We denote the first integral by I and the second integral by I . We now compute I using the definition of A l as I = n − m X l =0 i l +1 n +1 X α =0 ( − α a ( m, l, α ) Z ∞ Z g α ( T β ( x, t )) t n − m + s dxdt. Using the Fourier expansion for the α -th component of f , see [Gha99, (7)] with a = 1,we get f α ( T β ( x, t )) = t (cid:18) n + 2 α (cid:19) X ξ ∈ F × c ( ξd ) (cid:18) ξi | ξ | (cid:19) n +1 − α K α − n − (4 πt | ξ | ) e F ( ξ ( x + β )) , where e F ( w ) = e πi Tr F/ Q ( w ) . Using (8) and plugging this into the expression for I , we get I = n − m X l =0 i l +1 n X α =0 ( − α a ( m, l, α ) Z ∞ X ξ ∈ F × c ( ξd ) t n − m +1+ s (cid:18) ξi | ξ | (cid:19) n +1 − α K α − n − (4 πt | ξ | ) + ( − n + m +1 − α (cid:18) ξi | ξ | (cid:19) α − n − K n +1 − α (4 πt | ξ | ) ! dt Z e F ( ξ ( x + β )) dx + n − m X l =0 i l +1 ( − n +1 a ( m, l, n + 1) Z ∞ X ξ ∈ F × c ( ξd ) t n +1 − m + s K (4 πt | ξ | ) dt Z e F ( ξ ( x + β )) dx. The only terms c ( ξd ) that survive are when ξ = r √− D , for some 0 = r ∈ Z , and in thiscase R e F ( ξx ) dx = 1. I = n − m X l =0 i l +1 n X α =0 ( − α a ( m, l, α ) X r =0 e F ( rβ/ √− D ) c ( r ) (cid:18) − r | r | (cid:19) n +1 − α Z ∞ t n +1 − m + s (cid:20) K α − n − (cid:18) πt | r |√ D (cid:19) + ( − n + m +1 − α K n +1 − α (cid:18) πt | r |√ D (cid:19)(cid:21) dt + n − m X l =0 i l +1 ( − n +1 a ( m, l, n + 1) X r =0 e F ( rβ/ √− D ) c ( r ) Z ∞ t n +1 − m + s K (cid:18) πt | r |√ D (cid:19) dt. The Bessel functions have the property [Gha99, Lemma 7] Z ∞ K ν ( at ) t µ − dt = 2 µ − a − µ Γ (cid:18) µ + ν (cid:19) Γ (cid:18) µ − ν (cid:19) . This implies that the two Bessel functions in the sum above will cancel each other unless α ≡ n + 1 + m mod (2). Setting s ′ = 2 n + 2 − m + s , we have I = ( − n +1 √ D s ′ π ) n +2 − m + s n − m X l =0 i l +1 n X α =0 α ≡ n +1+ m (2) ( − m a ( m, l, α ) X = r ∈ Z e F ( rβ/ √− D ) c ( r ) (cid:18) − r | r | (cid:19) n +1 − α | r | s ′ Γ (cid:18) n + 1 − m + α + s (cid:19) Γ (cid:18) n + 3 − m − α + s (cid:19) + ( − n +1 √ D s ′ π ) n +2 − m + s n − m X l =0 i l +1 a ( m, l, n + 1) X = r ∈ Z e F ( rβ/ √− D ) c ( r ) 1 | r | s ′ Γ (cid:18) n + 2 − m + s (cid:19) . We will take β = b √− D for some rational number b . Then the term e F ( rβ/ √− D ) = e πirb .Now we break the sum over r into a sum over positive integers and a sum over negativeintegers. The term (cid:16) − r | r | (cid:17) n +1 − α equals ( − m when r is positive and is 1 when r is negative.The second sum over r does not have such a term, so we assume that m is even in orderto be able to put these terms together into a single term. The terms c ( r ) and | r | s ′ are -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 11 obviously independent of the sign of r . So finally, we have I = ( − n +1 √ D s ′ π ) n +2 − m + s ∞ X r =1 (cid:0) e πirb + e − πirb (cid:1) c ( r ) r s ′ n − m X l =0 i l +1 n +1 X α =0 α ≡ n +1+ m (2) a ( m, l, α )Γ (cid:18) n + 1 − m + α + s (cid:19) Γ (cid:18) n + 3 − m − α + s (cid:19) , (12)where there is an extra factor of in the α = n + 1 term, which we will adjust for. Bya similar computation, I will also have an expression in terms of b ( m, l, α ). Puttingtogether these two expressions, we get that h T ∗ β ( δ n − m ( f )) , E β n − m +2 ( s ) i = √ D s ′ (2 π ) n +2 − m + s ∞ X r =1 (cid:0) e πirb + e − πirb (cid:1) c ( r ) r s ′ · G ′∞ ( s, f ) , where we collect all the combinations of Gamma factors appearing in both I and I anddenote it by G ′∞ ( s, f ).Now let χ : ( Z /p j Z ) × → C × be a primitive character (so j = j χ ) and recall that G ( s, ¯ χ, f ) = L N (2 s − k + 2 , ¯ χ ) ∞ X r =1 c ( r ) ¯ χ ( r ) r s . Substituting the formula ¯ χ ( r ) = 1 G ( χ ) X a mod p j χ ( a ) e πira/p j in the above equation, we get G ( s ′ , ¯ χ, f ) = L N (2 n − m + 2 + 2 s, ¯ χ ) 1 G ( χ ) X a mod p j χ ( a ) ∞ X r =1 c ( r ) r s ′ e πira/p j . Now assume that χ is an even character, i.e., χ ( −
1) = 1. Then grouping together theterms coming from a and − a , we get G ( s ′ , ¯ χ, f ) = L N (2 n − m + 2 + 2 s, ¯ χ ) 1 G ( χ ) X a ∈ R χ ( a ) ∞ X r =1 c ( r ) r s ′ ( e πira/p j + e − πira/p j ) , where R is half of the representatives modulo p j such that if a ∈ R , then − a R . Wenow write G ( s ′ , χ, f ) in terms of the inner product considered earlier(13) G ( χ ) G ( s ′ , ¯ χ, f ) = (2 π ) n +2 − m + s G ′∞ ( s, f ) √ D s ′ L N (2 n − m + 2 + 2 s, ¯ χ ) X a ∈ R χ ( a ) h T ∗ β ( δ n − m ( f )) , E β n − m +2 ( s ) i , with β = a √− D/ p j .Let G ∞ ( s, f ) = G ′∞ ( s, f )Γ( s + 2 n − m + 2). Dividing both sides of the above equationby the period G ( ¯ χ )(2 π ) n − m +2 , we obtain G ( χ ) G ( s ′ , ¯ χ, f ) G ( ¯ χ )(2 π ) n − m +2 = (2 π ) n +2 − m + s G ∞ ( s, f ) √ D s ′ · L N (2 n − m + 2 + 2 s, ¯ χ )(2 π ) n − m +2 G ( ¯ χ ) · Γ( s + 2 n − m + 2) · X a ∈ R χ ( a ) h T ∗ β ( δ n − m ( f )) , E β n − m +2 ( s ) i , We evaluate this expression at s = 0. Note that G ∞ (0 , f ) = 0, by [Gha99, § L N (2 n − m + 2 , ¯ χ ) becomesrational after dividing by the period G ( ¯ χ )(2 π ) n − m +2 . We denote this ratio by L ◦ (2 n − m + 2 , ¯ χ ). We get(14) G ( χ ) G (2 n − m + 2 , ¯ χ, f ) G ( ¯ χ )Ω ∞ = L ◦ (2 n − m +2 , ¯ χ ) X a ∈ R χ ( a ) h T ∗ β ( δ n − m ( f )) , E β n − m +2 (0) i . Here Ω ∞ is defined as Ω ∞ = (2 π ) n − m +4 Γ(2 n − m + 2) G ∞ (0 , f ) √ D n − m +2 . We now conclude rationality properties of the special values G (2 n − m + 2 , ¯ χ, f )from equation (14). Choose a period Ω( f ) such that after dividing by this period, thedifferential form δ ◦ ( f ) := δ ( f )Ω( f ) ∈ H cusp (Γ ( n ) \H , L ( n, E ))takes rational values. Here E is a sufficiently large p -adic field, containing the field ofrationality of the form f , which we also view as a subfield of C after fixing an isomorphismbetween C and Q p . Then T ∗ β δ ( f ) | H = Ω( f ) · T ∗ β δ ◦ ( f ) | H , noting that if √− D ∈ E , which we assume, then the image T ∗ β δ ◦ ( f ) | H of δ ◦ ( f ) under themap T ∗ β | H : H cusp (Γ ( n ) \H , L ( n, E )) → H cusp (Γ β ( n ) \H , L ( n, E )) → H cusp (Γ β ( N ) \ H , L ( n, E )) , is also rational. Since Clebsch-Gordan preserves rationality, for 0 ≤ m ≤ n , we obtainthat T ∗ β ( δ n − m ( f )) = Ω( f ) · T ∗ β ( δ ◦ n − m ( f )) , where T ∗ β ( δ ◦ n − m ( f )) ∈ H cusp (Γ β ( N ) \ H , L (2 n − m, E )) is also rational.The rational cuspidal class T ∗ β ( δ ◦ n − m ( f )) is cohomologous to a compactly supportedrational class which has the same value when paired with E β n − m +2 (0) (see the proof of[Gha99, Theorem 1]). Since the differential form E β n − m +2 (0) coming from the Eisensteinseries is E -rational, at least when m = n (see Proposition 1 in Section 4 below), andthe pairing between compactly supported rational classes and such classes preserves E -rationality, the following theorem follows from (14), if E contains the field of rationalityof χ , which we again assume. Theorem 1 (Rationality result for twisted Asai L -values) . Let E be a sufficiently large p -adic number field with p ∤ N D . Let ≤ m < n be even and χ be even. Then G ( χ ) G (2 n − m + 2 , ¯ χ, f ) G ( ¯ χ )Ω( f )Ω ∞ ∈ E. This result matches with [Gha99, Theorem 1] when χ is trivial. In that theorem it wasassumed that the finite part of the central character of f is non-trivial primarily to dealwith the rationality of the Eisenstein series when m = n . In this paper, we have assumed(for simplicity) that the finite part of the central character of f is trivial. We could stillprobably include the case m = n in the theorem above, by using the rationality of theEisenstein series E β (0 , z ) − pE β (0 , pz ) instead (see [Gha99, Remark 2]). -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 13 Rationality of Eisenstein cohomology classes
We start by recalling the following result that goes back to Harder [Har81], [Har87].See also [Hid94, § Lemma 1.
Eisenstein cohomology classes corresponding to Eisenstein series whose con-stant terms at every cusp are rational are rational cohomology classes.Proof.
We use notation in this proof that is independent of the rest of the paper. LetΓ ⊂ SL ( Z ) be a congruence subgroup and L ( n, C ) denote the sheaf of locally constantsections of π : Γ \ ( H × L ( n, C )) → Γ \ H . Consider the restriction map to boundarycohomology given by R : H (Γ \ H , L ( n, C )) → H ∂ (Γ \ H , L ( n, C )) := M ξ H (Γ ξ \ H , L ( n, C )) , where ξ varies through the cusps of Γ. We know that H (Γ \ H , L ( n, C )) = H (Γ \ H , L ( n, C )) ⊕ H (Γ \ H , L ( n, C )) , where H and H are the cuspidal and Eisenstein part of cohomology respectively.The restriction of R to H is an isomorphism R : H (Γ \ H , L ( n, C )) → M ξ H (Γ ξ \ H , L ( n, C )) ω c ξ (0 , ω ) , where c ξ (0 , ω ) is the differential form corresponding to the “constant term” in the Fourierexpansion at the cusp ξ of the differential form ω corresponding to the underlying Eisen-stein series. Clearly R preserves the rational structures on both sides. The following factis due to Harder. Fact : There exists a section M : ⊕ ξ H (Γ ξ \ H , L ( n, C )) → H (Γ \ H , L ( n, C ))of R preserving rational structures on both sides.Now let ω ∈ H (Γ \ H , L ( n, C )) be such that R ( ω ) is rational. Then M ( R ( ω )) is rationaland R ( M ( R ( ω ))) = R ( ω ). Since R is an isomorphism we have M ( R ( ω )) = ω . Hence, ω is rational. Thus the Eisenstein class ω is rational if and only if the constant term in theFourier expansion at every cusp is rational. This proves the lemma. (cid:3) Proposition 1. If m = n , then the Eisenstein differential form E β n − m +2 (0) ∈ H (Γ β ( N ) \ H , L (2 n − m, E )) is rational, for a sufficiently large p -adic number field E .Proof. Recall that β = a √− D p j if j ≥ β = 0 if j = 0). We claim that Γ β ( N ) = γ − β Γ ( N ) γ β ∩ SL ( Z ) is independent of a . We do this by showing that(15) Γ β ( N ) = (cid:8) ( a bc d ) ∈ SL ( Z ) : a ≡ d mod p j , c ≡ N p j (cid:9) . Indeed, if j = 0, (15) holds trivially, since in this case γ β = 1, so both sides of (15)are equal to Γ ( N ). So assume that j ≥
1. Since p is odd and we are consideringrepresentatives a ∈ R = ( Z /p j Z ) × / {± } , by replacing a by p j − a if necessary, we may assume that all a ∈ R are even, so that p j √− D − β ∈ Z . Let γ = ( a bc d ) ∈ Γ ( N ) and γ β = (cid:0) β (cid:1) . Then(16) γ − β γγ β = (cid:18) a − cβ b − dβ + ( a − cβ ) βc d + cβ (cid:19) . Assume that the matrix in (16) is in SL ( Z ), so is in Γ β ( N ). Then a − cβ , b − dβ +( a − cβ ) β , c , d + cβ ∈ Z . Note that c ∈ Z ⇔ c ∈ N Z . Since a − cβ and d + cβ ∈ Z , we have b − dβ + ( a − cβ ) β ∈ Z ⇔ ( b + cβ ) + ( a − cβ − d − cβ ) β ∈ Z ⇔ ℜ ( b ) + cβ ∈ Z and ℑ ( b ) = i ( a − cβ − d − cβ ) β ⇔ p j | c and a − cβ ≡ d + cβ mod p j , since p ∤ D and both ℜ ( b ), √ D − ℑ ( b ) ∈ Z . Therefore Γ β ( N ) is contained in the righthand side of (15). On the other hand, if γ is any matrix on the right hand side of (15),then by replacing β by − β in (16), one checks that γ β γγ − β ∈ Γ ( N ). It follows thatequality holds in (15).From (15), we have SL ( Z ) ∞ = (cid:8)(cid:0) ± n ± (cid:1) : n ∈ Z (cid:9) ⊂ Γ β ( N ) . Thus Γ β ( N ) ∞ = SL ( Z ) ∞ . Also, note that the coset Γ β ( N ) ∞ ( a bc d ) in Γ β ( N ) ∞ \ Γ β ( N ) con-tains all the matrices of Γ β ( N ) whose bottom row equals ± ( c, d ). Hence Γ β ( N ) ∞ \ Γ β ( N )is in bijection with the setΛ := { ( c, d ) ∈ Z r { (0 , } : ( c, d ) = 1 , c ≡ N p j , d ≡ ± p j } / {± } . For each integer k ≥ u, v ) ∈ ( Z /N p j ) , consider the Eisenstein series E ( u,v ) k ( z ) := X ( c,d ) ≡ ( u,v ) mod Np j ( c,d )=1 cz + d ) k . This Eisenstein series differs from the Eisenstein series in [DS05, (4.4)] by a factor of ǫ Np j = or 1. By (11), we have E β n − m +2 (0 , z ) = X ( c,d ) ∈ Λ cz + d ) n − m +2 · ω = 12 X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j E ( u,v )2 n − m +2 ( z ) · ω, (17)noting that 2 n − m + 2 ≥
4, since m = n . By [DS05, (4.6)], for k ≥
3, we have E ( u,v ) k ( z ) = X l ∈ ( Z /Np j Z ) × ζ l + ( k, µ ) G l − ( u,v ) k ( z ) , (18)where ζ l + ( k, µ ) := ∞ X m =1 m ≡ l mod Np j µ ( m ) m k , -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 15 µ ( · ) is the M¨obius function, and G ( u,v ) k ( z ) := X ′ ( c,d ) ∈ Z ( c,d ) ≡ ( u,v ) mod Np j cz + d ) − k . We will obtain the q -expansion of the Eisenstein series E β n − m +2 (0 , z ) = P ∞ n =0 a n q n , using(17), (18) and the q -expansion of the Eisenstein series above using facts from [DS05].Let k ≥ ϕ be the Euler totient function. For v ∈ ( Z /N p j ) × , we have(19) ζ v ( k ) := X ′ d ≡ v mod Np j d − k = ∞ X d =1 d ≡ v mod Np j d − k + ( − k ∞ X d =1 d ≡− v mod Np j d − k = 1 ϕ ( N p j ) X ψ mod Np j ψ ( v ) − L ( k, ψ ) + ( − k X ψ mod Np j ψ ( − v ) − L ( k, ψ ) = 1 ϕ ( N p j ) X ψ mod Np j (1 + ( − k ψ ( − ψ ( v ) − L ( k, ψ ) , where the penultimate step follows from [DS05, Page 122]. If k is even, then (1 +( − k ψ ( − ψ is even (resp. odd). A similar expressionfor ζ l + ( k, µ ) in terms of Dirichlet L -functions can also be derived. For l ∈ ( Z /N p j Z ) × ,by the orthogonality relations, we have ζ l + ( k, µ ) = ∞ X m =1 ϕ ( N p j ) X ψ mod Np j ψ ( l ) − ψ ( m ) µ ( m ) m − k = 1 ϕ ( N p j ) X ψ mod Np j ψ ( l ) − ∞ X m =1 ψ ( m ) µ ( m ) m − k = 1 ϕ ( N p j ) X ψ mod Np j ψ ( l ) − L ( k, ψ ) − , (20)where the last step follows from by multiplying the corresponding L -functions. Therefore ζ l + ( k, µ ) + ζ − l + ( k, µ ) = 1 ϕ ( N p j ) X ψ mod Np j ( ψ ( l ) − + ψ ( − l ) − ) L ( k, ψ ) − = 2 ϕ ( N p j ) X ψ even ψ mod Np j ψ ( l ) − L ( k, ψ ) − . (21)By (17) and (18), we have E β n − m +2 (0 , z ) = 12 X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X l ∈ ( Z /Np j Z ) × ζ l + (2 n − m + 2 , µ ) G l − ( u,v )2 n − m +2 ( z ) · ω. (22)For simplicity, let k = 2 n − m + 2. By the description of the set Λ, we have ( c, d ) ∈ Λimplies that
N p j | c and ( c, d ) = 1, so the congruence class v of d mod N p j has order N p j . Therefore, for m < n , by [DS05, Theorem 4.2.3], we have G ( u ′ ,v ′ ) k ( z ) = ζ v ′ ( k ) + ( − πi ) k ( k − N p j ) k ∞ X l =1 σ ( u ′ ,v ′ ) k − ( l ) e πilz/Np j , (23)for tuples ( u ′ , v ′ ) occurring in (22), where σ ( u ′ ,v ′ ) k − ( l ) = X l ′ | ll/l ′ ≡ Np j sgn( l ′ ) l ′ k − e πiv ′ l ′ /Np j . Constant term:
By (22) and (23), the constant term a in the q -expansion of E βk (0 , z ) equals a = 12 X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X l ∈ ( Z /Np j Z ) × ζ l + ( k, µ ) ζ l − v ( k ) ( ∗ ) = X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X l ∈ ( Z /Np j Z ) × ϕ ( N p j ) X ψ mod Np j ψ ( l ) − L ( k, ψ ) − X ψ mod Np j ψ even ψ ( l − v ) − L ( k, ψ )= 1 ϕ ( N p j ) X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X ψ,ψ mod Np j ψ even ψ ( v ) − L ( k, ψ ) L ( k, ψ ) X l ∈ ( Z /Np j Z ) × ψ ψ − ( l ) ( ∗∗ ) = 1 ϕ ( N p j ) X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X ψ mod Np j ψ even ψ ( v ) − = 12 ϕ ( N p j ) X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X ψ mod Np j ( ψ ( v ) − + ψ ( − v ) − ) ( ∗∗∗ ) = 1 , where ( ∗ ) follows from (19) and (20), and ( ∗∗ ) and ( ∗ ∗ ∗ ) follow from the orthogonalityrelations. Higher Fourier coefficients:
Clearly σ (0 ,v ′ ) k − ( l ) = 0 if N p j ∤ l . So assume that l is a multiple of N p j . Say l = N p j l ′′ .Then σ (0 ,v ′ ) k − ( l ) = X l ′ | l ′′ sgn( l ′ ) l ′ k − e πiv ′ l ′ /Np j , which is clearly E -rational if E contains a sufficiently large cyclotomic number fielddepending on j . From (22) and (23), we see that the coefficient a l ′′ of q l ′′ in the Fourierexpansion of E βk (0 , z ) equals a l ′′ = 12 X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡± p j X n ∈ ( Z /Np j Z ) × ζ n + ( k, µ ) ( − πi ) k ( k − N p j ) k σ n − (0 ,v ) k − ( N p j l ′′ ) . -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 17 If j = 0, one checks that the formula for a l ′′ above reduces to a well-known expression (see[Miy89, Theorem 7.1.3 and (7.1.30)]), and in particular a l ′′ ∈ Q is rational. So assumethat j >
0. Then a l ′′ = 12 X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡ p j X n ∈ ( Z /Np j Z ) × ( ζ n + ( k, µ ) + ζ − n + ( k, µ )) ( − πi ) k ( k − N p j ) k σ n − (0 ,v ) k − ( N p j l ′′ ) (21) = 1 ϕ ( N p j ) X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡ p j X n ∈ ( Z /Np j Z ) × X ψ mod Np j ψ even ψ ( n ) − L ( k, ψ ) − ( − πi ) k ( k − N p j ) k · σ n − (0 ,v ) k − ( N p j l ′′ ) (24) = − kϕ ( N p j ) X ( u,v ) ∈ ( Z /Np j Z ) u ≡ Np j v ≡ p j X n ∈ ( Z /Np j Z ) × X ψ mod Np j ψ even ψ ( n ) − (cid:18) C ψ N p j (cid:19) k G ( ψ ) B k, ¯ ψ · σ n − (0 ,v ) k − ( N p j l ′′ ) , where in the last step we have used the following special value result for the Dirichlet L -function: L ( k, ψ ) = − ( − πi ) k G ( ψ ) B k, ¯ ψ k ! C kψ if ψ is even and k > , (24)where C ψ denotes the conductor of ψ . Thus a l ′′ is again E -rational for a sufficiently large p -adic number field E containing an appropriate cyclotomic number field.Summarizing, the computations above show that E β n − m +2 (0 , z ) has an E -rational q -expansion P ∞ n =0 a n q n (at the cusp ∞ ) if E contains a sufficiently large cyclotomic numberfield (which depends on j ). By [DS05, Proposition 4.2.1], since E ( u,v )2 n − m +2 | γ = E ( u,v ) γ n − m +2 ,for all γ ∈ SL ( Z ), the Eisenstein series E β n − m +2 (0 , z ) has an E -rational q -expansion ateach cusp ξ of Γ β ( N ). The proposition now follows from Lemma 1. (cid:3) Towards integrality
Note that the map T ∗ β | H can also be described as the pull-back of a differential formvia the map S β : Γ β ( N ) \ H → Γ ( n ) \H given by sending x + it γ β (cid:20) x − tt x (cid:21) . We now choose δ ◦ ( f ) such that it generates ¯ H cusp (Γ ( n ) \H , L ( n, O E ))[ f ], which is arank one O E -submodule of ¯ H cusp (Γ ( n ) \H , L ( n, O E )), where O E is the valuation ring of E and ¯ H denotes the image of the integral cohomology in the rational cohomology underthe natural map. We correspondingly refine the period Ω( f ) so that Ω( f ) ∈ C × / O × E .Since β = a √− D p j , we have γ − β · P ∈ L ( n, p nj O E ), for P ∈ L ( n, O E ). Thus, the map S β does not preserve cohomology with integral coefficients, but instead induces a map(25) S ∗ β : ¯ H cusp (Γ ( n ) \H , L ( n, O E )) → ¯ H cusp (Γ β ( N ) \ H , L ( n, p nj O E )) , on cohomology. Lemma 2.
Assume p > n . Then under the Clebsch-Gordan decomposition (7) , we have S ∗ β ( ¯ H cusp (Γ ( n ) \H , L ( n, O E )) n M m =0 ¯ H cusp (Γ β ( N ) \ H , L (2 n − m, p j (2 n − m ) O E )) . Proof.
Let ∇ = (cid:18) ∂ ∂X∂Y − ∂ ∂X∂Y (cid:19) . By [Gha99, Lemma 2], the projection to the m -thcomponent in (7) is induced by P ( X, Y, X, Y ) m ! ∇ m P ( X, Y, X, Y ) | X = XY = Y . Clearly theprojection continues to be defined with O E coefficients if p > n . As remarked in (25), S ∗ β does not preserve integrality. However, since ∇ ( γ − β · X n − k Y k X n − l Y l ) = l ∂∂X X n − k ( Y − βX ) k X n − l ( Y + βX ) l − − k ∂∂X X n − k ( Y − βX ) k − X n − l ( Y + βX ) l , we see that if P ∈ L ( n, O E ), the total power of p j in the denominator goes down by oneafter applying ∇ to γ − β · P . Iterating this, we see ∇ m ( γ − β · P ) ∈ L (2 n − m, p j (2 n − m ) O E ),for m = 0 , . . . , n , proving the lemma. (cid:3) We now assume that the prime p is greater than n , so that we may apply the lemmaabove. Let S ∗ β ( δ ◦ n − m ( f )) ∈ ¯ H cusp (Γ β ( N ) \ H , L (2 n − m, p j (2 n − m ) O E ))(26)be the image of δ ◦ ( f ) under the map (25) followed by the projection to the m -th compo-nent in the Clebsch-Gordan decomposition in Lemma 2. Again note the slight abuse ofnotation, since the subscript 2 n − m should be outside the brackets.By Proposition 1, we know that E β n − m +2 (0) ∈ p c j ¯ H (Γ β ( N ) \ H , L (2 n − m, O E )) , (27)for some integer c j ≥
0, depending on j .Let S denote the finite set of excluded primes above (i.e., p | N D and p ≤ n ), whichwe extend to include the primes p < n + 4. We remark that if p S , then p > n whichensures that the duality pairing h , i is a well-defined pairing on cohomology with integralcoefficients L (2 n − m, O E ).For the refined period Ω( f ) defined above, we get the following partial integrality result. Proposition 2.
Suppose p is not in the finite set of primes S , and that E is a sufficientlylarge p -adic number field as above. Let ≤ m < n be even and χ be an even character ofconductor p j χ . Then (28) G ( χ ) G (2 n − m + 2 , ¯ χ, f )Ω( f ) G ( ¯ χ )Ω ∞ ∈ O E p j χ (4 n − m +3)+ c jχ . Proof.
Indeed, this follows from the fact that by (14) the special value in the statementof the proposition has a cohomological description in terms of integrals of the form Z [Γ β ( N ) \ H ] S ∗ β δ ◦ n − m ( f ) ∧ E β n − m +2 . These are integrals of cohomology classes with specifiable denominator over an integralcycle, hence belong to O E with specifiable denominator. The size of the denominator canbe computed from (26) and (27), taking j = j χ , and the fact that the Dirichlet L -valuein (14) satisfies L ◦ (2 n − m + 2 , ¯ χ ) ∈ p jχ (2 n − m +3) O E (which in turn follows easily from a -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 19 special values result like (24), noting that the corresponding twisted Bernoulli numbers liein p jχ O E by a standard formula for these numbers involving the usual Bernoulli numbersup to B n +2 , and by the well-known result of von Staudt-Clausen which says that p doesnot divide the denominators of these Bernoulli numbers since p − > n + 2, by thedefinition of the set S ). (cid:3) Constructing bounded distributions
Finally, we now define our p -adic distribution by the formula µ ◦ n − m +2 = 1Ω( f )Ω ∞ · µ n − m +2 . These distributions are certainly defined whenever 2 n − m + 2 ≥ k + 2 which is the sameas m ≤ n −
2, but may possibly be defined for all 0 ≤ m ≤ n , by analytic continuation.We wish to show that µ ◦ n − m +2 is a bounded distribution and hence a measure. To thisend we recall the notion of abstract Kummer congruences. Theorem 2 (Abstract Kummer congruences) . Let Y = Z × p , let O p be the ring of integersof C p and let { f i } be a collection of continuous functions in C ( Y, O p ) such that the C p -linear span of { f i } is dense in C ( Y, C p ) . Let { a i } be a system of elements with a i ∈ O p .Then the existence of an O p -valued measure µ on Y with the property Z Y f i dµ = a i is equivalent to the following congruences: for an arbitrary choice of elements b i ∈ C p almost all zero, and for n ≥ , we have X i b i f i ( y ) ∈ p n O p , for all y ∈ Y = ⇒ X i b i a i ∈ p n O p . We apply this theorem with f i the collection of Dirichlet characters χ of ( Z /p j Z ) × , forall j ≥
1, thought of as functions of Y = Z × p , and with a χ ∈ O p the values of µ ( χ ), for agiven C p -valued distribution µ on Y . To prove that µ is an O p -valued measure on Y , itsuffices to prove Kummer congruences of the more specialized form(29) X χ χ − ( a ) χ ( y ) ∈ p j − O p , for all y ∈ Y = ⇒ X χ χ − ( a ) µ ( χ ) ∈ p j − O p , where χ varies over all characters mod p j , for a fixed j ≥
1, and where the first congruencein (29) follows from the identity P χ χ − ( a ) χ = φ ( p j )11 a + p j Z p , for 11 a + p j Z p the characteristicfunction of the coset a + p j Z p ⊂ Z × p . Indeed, then the second congruence in (29) showsthat µ is O p -valued on 11 a + p j Z p , whence on all O p -valued step functions on Z p , whence onall O p -valued continuous functions on Y . Claim.
The Kummer congruences (29) hold for µ = µ ◦ n − m +2 , for m ≤ n − even. In order to prove this claim we must show that the second sum P χ χ − ( a ) µ ◦ n − m +2 ( χ )in (29) should firstly a) be integral and secondly b) be in p j − O p . Now (28) shows thatfor any even character χ and m even:(30) Z χ dµ ◦ n − m +2 = 2 p j χ (2 n − m +1) G ( ¯ χ ) κ j χ G ( χ ) G (2 n − m + 2 , ¯ χ, f )Ω( f ) G ( ¯ χ )Ω ∞ ∈ p j χ (2 n − m +2)+ c jχ O E , at least if κ is a unit, which we have assumed. For odd characters χ , the integral abovevanishes. Thus (30) shows that the second sum above is in p j (2 n − m +2)+ cj O p , with c j = max χ c j χ . This is still quite far from the integrality claimed in part a). Assuming thatpart a) holds, one must then further prove the congruence in part b).In any case, assuming the Claim, we have that µ ◦ n − m +2 is a measure, for 0 ≤ m ≤ n − x p : Z × p → O p be the usual embedding. We now wish to glue the measures µ ◦ n − m +2 ,for 0 ≤ m even, into one measure µ ◦ satisfying (see [CP89, Lemma 4.4], noting q ( V ) thereequals 1)(31) Z Z × p χ dµ ◦ = ( − m/ Z Z × p x mp χ dµ ◦ n − m +2 . To do this, we again appeal to the abstract Kummer congruences in the theorem above.For the f i , we consider a slightly larger class of functions than the Dirichlet characters χ above, namely those of the form x − mp · χ , for 0 ≤ m ≤ n , with m even. We set a m,χ = ( − m/ µ ◦ n − m +2 ( χ ) ∈ O p , which should be equal to µ ◦ ( x − mp χ ), by (31) above. Wenow assume that Claim.
The a m,χ satisfy the abstract Kummer congruences: X m,χ b m,χ ( x − mp χ )( y ) ∈ p j − O p , for all y ∈ Y = ⇒ X m,χ b m,χ a m,χ ∈ p j − O p . It would then follow from Theorem 2 that there is a measure µ ◦ such that (31) holds.Note that the Kummer congruences in the latter claim actually imply the ones in theformer claim for µ ◦ n − m +2 , by choosing the b m ′ ,χ = χ − ( a ) if m ′ = m , and b m ′ ,χ = 0 if m ′ = m . We expect that the proof of these Kummer congruences should be similar tothe Kummer congruences proved by Panchishkin in his construction of the p -adic Rankinproduct L -function attached to two cusp forms f and g , described in detail in [Pan88] (seealso [CP04], and [GV19] where a sign similar to the one occurring in (31) is corrected).Since µ ◦ and µ ◦ n +2 agree on a dense set of functions, namely all χ , the measure µ ◦ isjust the measure µ ◦ n +2 . We now define the p -adic Asai L -function as the Mellin transformof the measure µ ◦ = µ ◦ n +2 : L p ( χ ) = Z Z × p χ ( a ) dµ ◦ n +2 , for all χ : Z × p → C × p . Acknowledgements:
The first author was supported by SERB grants EMR/2016/000840and MTR/2017/000114. The second and third authors thank T.N. Venkataramana foruseful conversations. A version of this paper has existed since about 2016. Recently, Lo-effler and Williams [LW19] have announced a construction of the p -adic Asai L -functionattached to a Bianchi cusp form using a method involving Euler systems. References [CP89] John Coates and Bernadette Perrin-Riou. On p -adic L -functions attached to motives over Q .Algebraic number theory, Adv. Stud. Pure Math. (1989), Academic Press, 23–54,[CP04] Michel Courtieu and Alexei Panchishkin. Non-Archimedean L -functions and arithmetical Siegelmodular forms . Second edition. Lecture Notes in Mathematics , Springer-Verlag, Berlin(2004).[DS05] Fred Diamond and Jerry Shurman. A first course in modular forms . Graduate Texts in Math-ematics , Springer-Verlag, New York (2005).[Gha99] Eknath Ghate.
Critical values of the twisted tensor L -function in the imaginary quadratic case .Duke Math. J. (1999), no. 3, 595–638.[Gh99b] Eknath Ghate. Critical values of twisted tensor L-functions over CM-fields . Automorphic forms,automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. PureMath. , Part 1, Amer. Math. Soc., Providence, RI (1999), 87–109. -ADIC ASAI L -FUNCTIONS ATTACHED TO BIANCHI CUSP FORMS 21 [GV19] Eknath Ghate and Ravitheja Vangala. Non-Archimedean Rankin L -functions . RamanujanMath. Society, Lecture Note Series (2019), 29 pp., to appear.[Har81] G¨unter Harder. Period integrals of Eisenstein cohomology classes and special values of some L -functions . Progr. Math. (1982), 103–142.[Har87] G¨unter Harder. Eisenstein cohomology of arithmetic groups. The case GL . Invent. Math. (1987), no. 1, 37–118.[Hid94] Haruzo Hida. On the critical values of L -functions of GL and GL × GL . Duke Math. J. (1994), no. 2, 431–529.[LSO14] Dominic Lanphier and Howard Skogman. Values of twisted tensor L -functions of automorphicforms over imaginary quadratic fields . With an appendix by Hiroyuki Ochiai. Canad. J. Math. (2014), no. 5, 1078–1109.[LW19] David Loeffler and Chris Williams. P -adic Asai L -functions of Bianchi modular forms .https://arxiv.org/abs/1802.08207 (2019), 34 pp.[Miy89] Toshitsune Miyake. Modular forms . Springer-Verlag, Berlin (1989).[Pan88] Alexei Panchishkin.
Non-Archimedean Rankin L -functions and their functional equations . Izv.Akad. Nauk SSSR Ser. Math. (1988), no. 2, 336–354. Indian Institute of Science Education and Research, Pashan, Pune 411021, India.
E-mail address : [email protected] School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai 400005, India.
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