On a Rankin-Selberg 𝐿-Function over Different Fields

Abstract

Given two unitary automorphic cuspidal representations and defined on and , respectively, with and being Galois extensions of , we consider two generalized Rankin-Selberg -functions obtained by forcefully factoring   and  . We prove the absolute convergence of these -functions for . 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 and , or “upstairs” in some extension field containing the composite extension . We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields and are relatively prime, the two different definitions give the same generating function.

1. Introduction

The generating function attached to pairs of automorphic representations has its origins in the papers of Rankin [1] and Selberg [2, 3]. As a consequence of the knowledge of the location and multiplicities of the poles of these -functions, they obtained nonvanishing results on the edge of the critical strip for Hecke’s -functions and asymptotic estimates for the growth of Fourier coefficients of modular forms. Moreover, considering the adelic setting in [4], Rankin-Selberg convolutions were applied to obtain more precise information regarding constant terms of Eisenstein series, and in [5] the analytic properties of Rankin-Selberg -functions were used to obtain multiplicity-one results. More recently, using a version of Selberg orthogonality [6], Liu and Ye computed the -level correlation function of high nontrivial zeros of automorphic -functions (see [7–9]) to give insight into the distribution of prime numbers. In the classical setting as in [4, 10, 11] we are given a Galois extension of and two automorphic cuspidal representations with unitary central characters on and   on . Suppose for any finite place of we have the associated conjugacy classes given by the Langlands correspondence in determined by , and in associated with . Then the finite part of the Rankin-Selberg -function is given by the product of local factors where and denotes the cardinality of the residue field at a finite place of . We will always write , where lies over the prime in and denotes the modular degree (which will be the same for any lying over since is Galois) where denotes the ring of integers in with unique prime ideal . If a finite place of lies over the prime in then we will always denote  .

The Euler product in (3) is known to converge absolutely for (see [4]), and this result is crucial in the proofs of Theorems 1 and 2. In this paper we consider two possibly different Galois extensions and of of degrees and , respectively, with and given as before except that is defined on and is defined on . 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 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 for any then here means the place of lies over the prime in . 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 and on we say that 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 . 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 . Indeed, if and 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 with a unitary automorphic cuspidal on and a Galois extension of for . One possible strategy is to derive a prime number theorem by investigating the asymptotic behavior of the sum as , where 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 in [16] assuming is cyclic of prime degree for , thus generalizing the results of Liu and Ye [7–9]. A much more extensive goal would be to set 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 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 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 . We note that this result is unconditional.

Theorem 1. Let and be unitary automorphic cuspidal representations of and , respectively, with and being finite Galois extensions of . Then converges absolutely for .

Recall that given an extension of fields , and two automorphic representations and defined on and , respectively, we say is the base change lift of if for any finite place of lying over the place of we have where are the Satake parameters associated with at the place for . If the base change functor exists, we could just as well consider the convolution where denotes the base change functor for any extension , 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 and denote the local rings with unique maximal ideals and 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 , then we define the Rankin-Selberg -function we again obtain absolute convergence for .

Theorem 2. Let and be finite Galois extensions of of degrees and , and let and be unitary automorphic cuspidal representations of and , respectively; then converges absolutely for .

We close by relating and when and obtain analytic continuation when and are the trivial representations.

Theorem 3. Let and be as in Theorems 2 and 1, and suppose that ; then Moreover, if and denote the trivial representations of and , respectively, then where is the Dedekind zeta function of the composite extension . The identity (22) also holds if we only assume that .

2. Convergence of the Euler Products

Proof of Theorem 1. First we expand the product Now using the bound for the local parameters as in [7] for and lying over the prime , and , then for sufficiently large we can expand into a geometric series to get 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 and the second factor is a subproduct of . Hence these both converge absolutely for .

Proof of Theorem 2. For sufficiently large we can expand the inner factor into a geometric series Now taking absolute values and applying Cauchy-Schwarz we get We can write the right-hand side of the inequality as Since is also a Galois extension of , 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 and similarly , and thus and . 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 and hence is absolutely convergent for .

3. Proof of Theorem 3

For any two Galois extensions and of 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 . Then since for any prime we have the identities and and since the degree of the composite extension is , we get that the modular degree of any prime lying in the ring of integers of 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 and since and we get since . Finally we have that and imply that since , so that . By a similar argument we get that ; thus, where the last equality follows from the expansion of as in the proof of Theorem 1. If , this is merely a restatement that the automorphic induction and base change functors are adjoint to one another (see Proposition   in [20] for a nice exposition).

Returning to the case of the Dedekind zeta functions, if we relax the condition that and assume that , 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 of the absolute Galois group , where we set for any number field , the induced representation, can be canonically identified with the group algebra of the finite group . Moreover, the factorization of corresponds to the decomposition of into irreducibles under the right regular representation. In particular, if we denote by the set of equivalence classes of irreducible representations of a finite group , then by [19] we can write where denotes the Artin -function attached to the irreducible representation . By abuse of notation, we write to mean that we are given an irreducible representation of the finite group and we obtain by composing with the projection so that . Similarly, write with for some irreducible representation of . Then the convolution may be rewritten as Note that since we have an isomorphism given by . The tensor product factors through the subgroup which gives a representation of . Again letting , we get the identity

On the other hand, suppose that we are given an irreducible representation of with corresponding space . Then we obtain an irreducible representation of the direct product . Thus there exist irreducible representations and of and , respectively, and a one-to-one and onto intertwining map such that for any For any finite prime we can choose an embedding which induces a homomorphism of Galois groups where for all . Moreover, we have a short exact sequence and is topologically generated by the Frobenius automorphism . In other words, where the closure takes place in the Krull topology. We choose which maps to and the local Artin -factor at is by definition where . Unraveling, this gives where we have written , . Note that we have the identity which follows since if and only if if and only if , which proves the claim. Thus we can write as but this is precisely . It follows that So we again obtain all the analytic properties for free.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.