THE LS METHOD FOR THE CLASSICAL GROUPS IN POSITIVE CHARACTERISTIC AND THE RIEMANN HYPOTHESIS

Abstract

Abstract. We provide a definition for an extended system of γ-factors for products of generic representations τ and pi of split classical groups or general linear groups over a non-archimedean local field of characteristic p. We prove that our γ-factors satisfy a list of axioms (under the assumption p 6 = 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, τ×pi), where τ 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. introduction Let Gm and Gn denote either split classical groups or general linear groups of ranks m and n, respectively. Let k be a global function field with finite field of constants Fq and ring of adèles Ak. We present a theory of automorphic L-function