The local converse theorem for SO(2n+1) and applications

In this paper we characterize irreducible generic representations of SO_2n+1(k) (where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of SO_2n+1(A) (where A is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of SO_2n+1(k).