Strong exponent bounds for the local Rankin-Selberg convolution

TL;DR

This paper establishes precise bounds for the Artin and Swan exponents of tensor products of Weil-Deligne representations, with implications for local Rankin-Selberg convolutions in number theory.

Contribution

It provides new strong bounds for exponents of tensor products of Weil-Deligne representations, enhancing understanding of local Rankin-Selberg convolutions.

Findings

Derived upper and lower bounds for Artin and Swan exponents.

Introduced bounds involving tensor products with dual representations.

Applied bounds to Rankin-Selberg exponents via Langlands correspondence.

Abstract

Let $F$ be a non-Archimedean locally compact field. Let $\sigma$ and $\tau$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $\sigma\otimes\tau$ in terms of those of $\sigma$ and $\tau$. We give a different lower bound in terms of $\sigma\otimes\check\sigma$ and $\tau\otimes\check\tau$. Using the Langlands correspondence, we obtain the bounds for Rankin-Selberg exponents.