Uniqueness of Rankin-Selberg products

TL;DR

This paper proves the equivalence of different definitions of gamma-factors for Rankin-Selberg products, extending known results to positive characteristic and establishing a refined local-global principle for GL_n.

Contribution

It demonstrates the equality of gamma-factors from different methods in positive characteristic, providing a new local-global principle for GL_n and extending Shahidi's results.

Findings

Equality of gamma-factors in positive characteristic

Refined local-global principle for GL_n

Extension of Shahidi's results to new cases

Abstract

In the present paper, we show the equality of the $\gamma$-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands-Shahidi method. Our results are new in the case of positive characteristic, where we establish a refined version of the local-global principle for ${\rm GL}_n$ which has independent interest. In characteristic zero, the results are due to Shahidi. The comparison of $\gamma$-factors is made via a uniqueness result for Rankin-Selberg $\gamma$-factors.