On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms

We prove an algebraicity result for the central critical value of certain Rankin–Selberg L-functions for GL_n × GL_n−1. This is a generalization and refinement of the results of Harder [14], Kazhdan, Mazur, and Schmidt [23], and Mahnkopf [29]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality, one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. These results, as in the above works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals

Eisenstein Cohomology and ratios of critical values of Rankin-Selberg L-functions

This is an announcement of results on rank-one Eisenstein cohomology of GL(N), with N > 1 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin-Selberg L-functions for GL(n) x GL(n') when n is even and n' is odd.Comment: Final version to appear in Comptes Rendus Math. Acad. Sci. Pari

Ratios of periods for tensor product motives

In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for ${\rm GL}_n \times {\rm GL}_{n'}$ proved by G\"unter Harder and the second author.Comment: In this revised version, we have made some minor modifications according to the comments by the refere

Conductors and newforms for U(1,1)

Let $F$ be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms for $U(1,1)(F)$, building on previous work on $SL_2(F)$. This theory is analogous to the results of Casselman for $GL_2(F)$ and Jacquet, Piatetski-Shapiro, and Shalika for $GL_n(F)$. To a representation $\pi$ of $U(1,1)(F)$, we attach an integer $c(\pi)$ called the conductor of $\pi$, which depends only on the $L$-packet $\Pi$ containing $\pi$. A newform is a vector in $\pi$ which is essentially fixed by a congruence subgroup of level $c(\pi)$. We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.Comment: 25 page