The LS method for the classical groups in positive characteristic and the Riemann Hypothesis

TL;DR

This paper extends the theory of gamma and L-factors for classical and general linear groups over non-archimedean fields of positive characteristic, proving their properties and satisfying the Riemann Hypothesis for automorphic L-functions.

Contribution

It introduces a new framework for gamma-factors and L-functions in positive characteristic, establishing their axioms, uniqueness, and the Riemann Hypothesis for automorphic L-functions.

Findings

Defined extended gamma-factors satisfying axioms

Constructed automorphic L-functions with functional equations

Proved the Riemann Hypothesis for these L-functions

Abstract

We provide a definition for an extended system of $\gamma$-factors for products of generic representations $\tau$ and $\pi$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We prove that our $\gamma$-factors satisfy a list of axioms (under the assumption $p \neq 2$ when both groups are classical groups) and show their uniqueness (in general). This allows us to define extended local $L$-functions and root numbers. We then obtain automorphic $L$-functions $L(s,\tau \times \pi)$, where $\tau$ and $\pi$ are globally generic cuspidal automorphic representations of split classical groups or general linear groups over a global function field. In addition to rationality and the functional equation, we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.