Arithmetic Study Group: Zeros of representation zeta functions
28 October 2014 14:00 in CM103
A representation zeta function \zeta_G(s) is a (meromorphic continuation of a) Dirichlet series whose nth coefficient r_n(G) counts the number of irreducible representations of dimension n of the group G, provided this number is finite. Currently little is known, or even conjectured, about the zeros of representation zeta functions. Very recently Kurokawa and Ochiai conjectured that if G is an infinite compact group such that r_n(G) is finite for all n, then the representation zeta function of G has a zero at -2. We will present a proof of this for compact p-adic analytic groups, due to Gonzalez-Sanchez, Jaikin-Zapirain and Klopsch. For finite groups it is a classical result that \zeta_G(-2) equals the order of the group, so this result can be interpreted as saying that the order of certain infinite groups is zero.
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