The LS method for the classical groups in positive characteristic and the riemann hypothesis

We provide a definition for an extended system of γ-factors for products of generic representations τ and π 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 ≠ 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 ×π), where τ and π 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.