Lethbridge Number Theory and Combinatorics Seminar: Qing Zhang

L-functions associated with automorphic forms are vast generalizations of Riemann zeta functions and Dirichlet L-functions. Although the theory of L-functions play a fundamental role in number theory, it is still largely conjectural. If π is an irreducible cuspidal automorphic representation of GLn over a number field F and π~ is its dual representation, it is conjectured that the Dedekind zeta functions ζF(s) (which is the Riemann zeta function when F=Q) "divides" the Rankin-Selberg L-function L(s,π×π~), i.e., the quotient L(s,π×π~)/ζF(s) (which is called the adjoint L-function of π) should be entire. For n=2, this conjecture was proved by Gelbart-Jacquet. In this talk, I will give a sketchy survey of constructions of some L-functions, including the Rankin-Selberg L-function L(s,π1×π2), and report our recent work on the above conjecture when n=3. This is a joint work with Joseph Hundley.