The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions

Abstract

We prove the nonvanishing hypothesis at infinity for Rankin-Selberg convolutions for GL(n) × GL(n − 1).2000 Mathematics Subject Classification. 22E41, 22E47.

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

“…Often times, a cohomological interpretation of an analytic theory of L-functions, depends on an assumption that a quantity coming from archimedean considerations in nonzero. In our situation, we adapt the methods of recent work of Sun [41] to prove an appropriate nonvanishing result; see Prop. 3.3.4.…”

On the nonvanishing hypothesis for Rankin-Selberg convolutions for 𝐺𝐿_{𝑛}(ℂ)×𝔾𝕃_{𝕟}(ℂ)

“…Let u 3 be a nonzero highest weight vector of β ∨ 3 , and let v 3 be a nonzero lowest weight vector of γ ∨ 3 . By Lemma 2.11 of [16]…”

“…The nonvanishing hypothesis is vital to the arithmetic study of critical values of higher degree L-functions and to the constructions of higher degree p-adic L-functions. Recently, Sun made a breakthrough by confirming it for GL n (R)×GL n−1 (R) and GL n (C)×GL n−1 (C), see [16]. The current paper aims to consider the GL n (C) × GL n (C) case, which has been expected by Grenié since 2003, see page 284 of [5].…”

“…For a general critical place j of π µ × π ν , we succeeded in using the translation functor (see Charper 7 of [10]) to realize I j as a submodule of I j 0 ⊗ F j with multiplicity one, where F j is certain finite dimensional representation of g. However, when j = j 0 , the lowest K-type of I j has multiplicity greater than one in I j 0 ⊗ F j . This prevents us from obtaining an analogue of Proposition 3.5 of [16].…”

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

“…Recent results on these lines are contained in [14,16,3]. The important paper of Binyong Sun [17], which proves the non-vanishing of the relevant archimedean zeta integrals, shows that the cup product expressions can be used effectively to relate the critical values to natural period invariants obtained by comparing the rational structures defined by Whittaker models to those defined cohomologically. There is no obvious relation, however, between these Whittaker periods and the motivic periods that enter into the computation of Deligne's period invariant c`.…”