Special values of $L$-functions and the refined Gan-Gross-Prasad conjecture

Abstract

We prove explicit rationality-results for Asai-L-functions, L S (s, Π ′ , As ± ), and Rankin-Selberg L-functions, L S (s, Π × Π ′ ), over arbitrary CM-fields F , relating critical values to explicit powers of (2πi). Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of (2πi), it is one of the crucial advantages of our refined approach, that it applies to very general non-cuspidal isobaric automorphic representations Π ′ of GLn(AF ). As a major application, this enables us to establish a certain algebraic version of the Gan-Gross-Prasad conjecture, as refined by N. Harris, for totally definite unitary groups: This generalizes a deep result of Zhang and complements totally recent progress of Beuzard-Plessis. As another application we obtain a generalization of an important result of Harder-Raghuram on quotients of consecutive critical values, proved by them for totally real fields, and achieved here for arbitrary CM-fields F and pairs (Π, Π ′ ) of relative rank one.

On algebraicity of special values of symmetric 4-th and 6-th power L-functions for $$\mathrm {GL}(2)$$

“…Moreover, we know that BC(Π 2 ) is cuspidal by [7,Theorem 7.3]. Now, we recall a result by Grobner-Lin [11], which is a key ingredient of our proof. Let κ µ (resp.…”

“…Hence, in this context, it is natural to assume that k has the same parity. On the other hand, in the proof of above theorems, we have to assume that π(f ) has trivial central character in order to apply Grobner-Lin [11] to certain automorphic representations (for example, see Theorem 7). Then this forces us to consider the case k i ≡ 0 (mod 2) for all i.…”

“…In the above three theorems, we only considered the algebraicity without Galois equivariance. As we noted, in our proof, we use several algebraicity results, and among of them, the algebraicity results by Grobner-Lin [11] and Liu [25] are crucial in our proof. In [11], they study an algebraicity of special values of automorphic L-functions for GL n (A E ) × GL n−1 (A E ) with a totally imaginary quadratic extension E of F , and they prove the Galois equivariance only for Gal(Q/E Gal ) with the Galois closure E Gal of E over Q.…”

“…As we noted, in our proof, we use several algebraicity results, and among of them, the algebraicity results by Grobner-Lin [11] and Liu [25] are crucial in our proof. In [11], they study an algebraicity of special values of automorphic L-functions for GL n (A E ) × GL n−1 (A E ) with a totally imaginary quadratic extension E of F , and they prove the Galois equivariance only for Gal(Q/E Gal ) with the Galois closure E Gal of E over Q. Moreover, in [25], she studies algebraicity of special values of standard L-functions of Siegel cuspforms and proves the Galois equivariance for ρ ∈ Gal(Q/Q) fixing certain Q(ζ N ).…”

“…Remark 11). Indeed, when F ̸ = Q, if we assume Galois equivariance for [11] and [25], we may prove the Galois equivariance for these theorems (see Section 2.5 and Remark 15). Further, we note that without any assumption, we can prove a partial Galois equivariance and it implies that the above values are in smaller fields in some special cases when F ̸ = Q (see Remark 14).…”