Whittaker Periods, Motivic Periods, and Special Values of Tensor Product -Functions

Abstract

Abstract. Let K be an imaginary quadratic field. Let Π and Π ′ be irreducible generic cohomological automorphic representation of GL(n)/K and GL(n − 1)/K, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if Π is cuspidal and the weights of Π and Π ′ are in a standard relative position, the critical values of the Rankin-Selberg product L(s, Π × Π ′ ) are essentially algebraic multiples of the product of the Whittaker periods of Π and Π ′ . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal Π can be given a motivic interpretation, and can also be related to a critical value of the adjoint L-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin-Selberg and adjoint L-functions are compatible with Deligne's conjecture. 1. Introduction L-functions can be attached both to automorphic representations and to arithmetic objects such as Galois representations or motives, and one implication of the Langlands program is that L-functions of the second kind are examples of L-functions of the first kind. Very few results of arithmetic interest can be proved about the second kind of L-functions until they have been identified with automorphic L-functions. For example, there is an extraordinarily deep web of conjectures relating the values at integer points of arithmetic (motivic) L-functions to cohomological invariants of the corresponding geometric (motivic) objects. In practically all the instances 1 where these conjectures have been proved, automorphic methods have proved indispensable. At the same time, there is a growing number of results on special values of automorphic L-functions that make no direct reference to arithmetic. Instead, the special values are written as algebraic multiples of complex invariants defined by means of representation theory. Examples relevant to the present paper ′ ) of a cuspdial automorphic representation Π of GL n and a cuspidal or abelian automorphic representation Π ′ of GL n−1 , where the general linear groups are over an imaginary quadratic field K. Here, the notion "abelian automorphic" refers to a representation, which is a tempered Eisenstein representation induced from a Borel subgroup of GL n−1 . In view of Raghuram's recent preprint [39], which we received after writing a first version of this paper, the inclusion of such automorphic representations Π ′ is the new feature of this result. The theorem applies in particular when Π and Π ′ are obtained by base change from cohomological cuspidal representations π and π ′ of unitary groups. As in [35], [40] and [39] the critical values of these L-functions are then expressed in terms of Whittaker periods, which are p...

Period Relations and Special Values of Rankin-Selberg L-Functions

“…Following [12], we define period invariants generalizing those introduced in [8]. Note that, in contrast to [8] and [3], it is not assumed here that M is polarized.…”

“…We have the following equation for the Deligne period with respect to the above bases: Since the previous proposition concerns only one place, we may assume that the base field F`" Q. In this case, the previous proposition was proved in [10,3] when the motive is polarized, and was generalized to non-polarized motives in [13,12]. Recall by definition that the image of p ω i in M DR b σ C is equal to I 8,σ pω i,σ q.…”

“…Condition (5.2) corresponds to the Good Position Hypothesis. When F is imaginary quadratic, the set of m satisfying the property (5.3) is identified in Lemma 3.5 of [3] with the set of critical points of the L-function Lps, Πb Π 1 q to the right of the center of symmetry of the functional equation (including the center if it is an integer). The same calculation holds for general CM fields.…”

p-Adic and analytic properties of period integrals and values of L-functions

“…Assuming a reasonable theory of motives, the article [16] gives precise conjectural versions of Principle 3.3 when F is an imaginary quadratic field. More general (hypothetical) factorizations are worked out in Section 4 of [11], and versions of these relations are proved in many cases in [11,12], and [24] A precise conjectural relation among the period invariants appearing on the left-hand and right-hand sides of (1.1) is given in Conjecture 5.16 of [15] (with the left-hand side correctly defined as in (3.1) above). In what follows, we will want to view algebraic numbers such as P(π, π ) −1 L can ( f, f ) as elements of p-adic fields (or integer rings), and to relate p-adic properties of these algebraic numbers to p-adic properties of the algebraic parts of the special values of L-functions on the right-hand side of (1.1).…”

“…The value at s = 1 of L(s, π, Ad) is critical and is therefore conjecturally an algebraic multiple of the corresponding Deligne period. (Automorphic versions of this conjecture are proved in many cases in [11,12].) I would like to say that the Bloch-Kato conjecture expresses the quotient of L(1, π, Ad) by the Deligne period in terms of orders of Galois cohomology groups.…”