On a Rankin-Selberg L-Function over Different Fields

m 󸀠 (A F ), respectively, with E and F being Galois extensions of Q, we consider two generalized Rankin-Selberg L-functions obtained by forcefully factoring L(s, π) and L(s, π 󸀠 ). We prove the absolute convergence of these L-functions for Re(s) > 1. The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension between E∩F andQ, or “upstairs” in some extension field containing the composite extension EF. We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields E and F are relatively prime, the two different definitions give the same generating function.


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Journal of Numbers different Galois extensions  and  of Q of degrees ℓ and ℓ  , respectively, with  and   given as before except that  is defined on   (A  ) and   is defined on    (A  ).For any prime number  and any place of  lying over  we denote by   the ramification index and   the number of places of  lying over the prime .Thus, we have ℓ =       and we similarly denote by    and    the ramification index and number of places of  lying over the prime .So the question arises: how to define a "Rankin-Selberg" -function attached to the pair (,   )?We begin by forcefully factoring the standard -functions of Jacquet [12]; that is, if we denote () =  2 for any  ∈ C then here ] |  means the place ] of  lies over the prime  in Z.
Similarly, if  denotes any finite place of  and    denotes the modular degree of any place of  lying over the prime  then Then we define the Rankin-Selberg -function as the following product: Note that Theorem 1 shows that this product is indeed well-defined.Recall from [13] that given two automorphic representations  on   (A  ) and Π on  ℓ (A Q ) we say that Π is automorphically induced from  if, for almost every prime number , This is well-defined by the strong multiplicity-one theorem, in which case we write  /Q () = Π.So if the automorphic induction functor exists for both the extensions  and  and the representations  and   , we could simply define where the -function on the right-hand side is the classical Rankin-Selberg -function of the pair ( /Q (),  /Q (  )).Indeed, if  /Q () and  /Q (  ) were known to exist globally, then (, × ,   ) would be a product of the classical Rankin-Selberg -functions as given in (3).The original motivation for considering such a convolution was to investigate the asymptotic behavior of the level correlation function (as in [14]) of high nontrivial zeros attached to a product  (,  1 )  (,  2 ) ⋅ ⋅ ⋅  (,   ) , (10) with   a unitary automorphic cuspidal on    (A   ) and   a Galois extension of Q for  = 1, 2, . . ., .One possible strategy is to derive a prime number theorem by investigating the asymptotic behavior of the sum as  → ∞, where Λ() denotes the Von Mangoldt function, and the coefficients  × ,   () come from the logarithmic derivative Once this is established one could ostensibly compute the leading order term of the -level correlation function.A prime number theorem was derived in the case where  and  are cyclic of prime degree in [15].Using this, the author computed the correlation function of a product (,  1 )(,  2 ) ⋅ ⋅ ⋅ (,   ) in [16] assuming   is cyclic of prime degree for  = 1, 2, . . ., , thus generalizing the results of Liu and Ye [7][8][9].A much more extensive goal would be to set  = Q and prove for a suitable collection of representations   that the -function (, × ,   ) enjoys "nice" analytic properties (e.g., holomorphic continuation, boundedness in vertical strips, and a functional equation); then it would follow from a converse theorem [17] that if  /Q () is suitably defined as an admissible representation, then it must be automorphic.
The main problem is that beyond the case when  is a cyclic extension of prime degree where the principle of functoriality holds as in [13] (or more generally a solvable extension built from towers of cyclic prime degree extensions), the properties of the global automorphic induction  /Q () are not well known (see [18]).Hence, unless the principle of functoriality as in [19] is established for any number field, the analytic properties of (, × ,   ) still need to be worked out.One approach to obtain the meromorphic continuation and functional equation would be the theory of Rankin-Selberg integrals following the work of Jacquet et al. [4].To show (, × ,   ) is bounded in vertical strips and to find the location of the poles one could try using the constant terms of Eisenstein series following the approach in [11].Establishing the meromorphic continuation is beyond reach at this point; nonetheless, we are able to show the absolute convergence of (, × ,   ) for Re() > 1.We note that this result is unconditional.Theorem 1.Let  and   be unitary automorphic cuspidal representations of   (A  ) and    (A  ), respectively, with  and  being finite Galois extensions of Q.Then (, × ,   ) converges absolutely for Re() > 1.
Recall that given an extension of fields  1 / 2 , and two automorphic representations   1 and   2 defined on   (A  1 ) and   (A  2 ), respectively, we say   1 is the base change lift of   2 if for any finite place ] 1 of  1 lying over the place ] 2 of  2 we have where     (, ]  ) are the Satake parameters associated with    at the place ]  for  = 1, . . ., .If the base change functor exists, we could just as well consider the convolution where   1 / 2 denotes the base change functor for any extension  1 / 2 , but again this assumes functoriality which only holds for a cyclic (or more generally solvable) extension.
In this vein let  denote any finite place of the composite extension , and if  lies over the place ] coming from the extension , let  |] denote the index by where O  and O ] denote the local rings with unique maximal ideals P  and P ] coming from the places  and ], respectively.Similarly, if  also lies over the place  coming from  let  | denote the index by Given  as before, define a new set of local parameters attached to the composite extension  for any place  of : where if  lies over ] Similarly, if  lies over the place  of  define If   denotes the cardinality of the residue field O  /P  , then we define the Rankin-Selberg -function we again obtain absolute convergence for Re() > 1.
Theorem 2. Let  and  be finite Galois extensions of Q of degrees ℓ and ℓ  , and let  and   be unitary automorphic cuspidal representations of   (A  ) and    (A  ), respectively; then (,   ×    ) converges absolutely for Re() > 1.
We close by relating (, × ,   ) and (,   ×    ) when (ℓ, ℓ  ) = 1 and obtain analytic continuation when  and   are the trivial representations.Theorem 3. Let  and   be as in Theorems 2 and 1, and suppose that (ℓ, ℓ  ) = 1; then Moreover, if 1  and 1  denote the trivial representations of   (A  ) and    (A  ), respectively, then where   () is the Dedekind zeta function of the composite extension .The identity (22) also holds if we only assume that  ∩  = Q.

Convergence of the Euler Products
Proof of Theorem 1.First we expand the product (1 −   (, ]) 1/     (, ) Now using the bound for the local parameters as in [7] for ] and  lying over the prime , |  (, ])| ≤    (1/2−1/(ℓ 2 +1)) and , then for Re() =  sufficiently large we can expand into a geometric series to get ) . (24) Thus by the triangle and Cauchy-Schwarz inequalities we put The first sum over  may be considered as a sum over the multiples of    , and similarly the sum over  may be considered as a sum over multiples of   .Thus we get the inequality The first factor above is a subproduct of the classical Rankin-Selberg -function (,  × π) and the second factor is a subproduct of (,   × π ).Hence these both converge absolutely for  > 1.
Proof of Theorem 2. For Re() =  sufficiently large we can expand the inner factor into a geometric series We can write the right-hand side of the inequality as )) )) Since  is also a Galois extension of Q, we have that / and / are likewise Galois, and so the numbers  |] and  | depend only on ] and , respectively.So we may write  |] =  ] and  | =   .Let  ] and   denote the number of places in  lying over the places ] and , respectively.Note that since we get [ : ] ≤ ℓ  and similarly [ : ] ≤ ℓ, and thus  ] ≤ ℓ  and   ≤ ℓ.So we can rewrite the above product as )) By the same argument as in Theorem 1 we get the inequality )) )) This last product is a subproduct of (,  × π) ℓ  /2 (,   × π ) ℓ/2 and hence is absolutely convergent for  > 1.

Proof of Theorem 3
For any two Galois extensions  and  of Q consider the Dedekind zeta functions Taking the convolution and expanding we get the Euler product The most tractable case is when the degrees of the extensions are relatively prime, so first assume that (ℓ, ℓ  ) = 1.Then since for any prime  we have the identities ℓ =       and ℓ  =       −   and since the degree of the composite extension /Q is ℓℓ  , we get that the modular degree of any prime lying in the ring of integers of /Q over  is given by      .Moreover, the number of prime ideals in the integral closure of  lying over  is      .Thus we get that the above Euler product is the Dedekind zeta function of the composite Hence we obtain analytic continuation and a functional equation for free in this case.Going back to the notation of Theorem 2 we also have that where the last equality follows from the expansion of (, × ,   ) as in the proof of Theorem 1.If  = Q, this is merely a restatement that the automorphic induction and base change functors are adjoint to one another (see Proposition 2.3.1 in [20] for a nice exposition).
Returning to the case of the Dedekind zeta functions, if we relax the condition that (ℓ, ℓ  ) = 1 and assume that ∩ = Q, then the situation requires more work.Since Langlands functoriality principle as in [19] follows easily, we may work with the associated Artin -function.Indeed, since   () is the Artin -function attached to the trivial representation 1  of the absolute Galois group Γ  , where we set Γ  = Gal(Q/) for any number field , the induced representation, can be canonically identified with the group algebra C[Γ /Q ] of the finite group Γ /Q = Γ Q /Γ  = Gal(/Q).Moreover, the factorization of   () corresponds to the decomposition of C[Γ /Q ] into irreducibles under the right regular representation.In particular, if we denote by Ĝ the set of equivalence classes of irreducible representations of a finite group , then by [19] we can write   () =  (, Ind