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

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.