Rationality of representation zeta functions of compact p-adic analytic groups

A representation zeta function of a group G is a (meromorphic continuation of) a Dirichlet series in a complex variable s whose n-th coefficient is the number of irreducible representations of dimension n of G (supposing that these numbers are finite). In 2006 Jaikin-Zapirain proved one of the most fundamental results in the area, namely that if G is a FAb compact p-adic analytic group (e.g., SL_n(Z_p)) and p > 2, then the representation zeta function of G is "virtually rational" in p^{-s}. Two reasons why such a result is interesting is that it immediately implies meromorphic continuation of the zeta function and that its abscissa of convergence is a rational number.

In the talk, I will explain what FAb and "virtually rational" mean here and outline recent joint work with M. Zordan on a new proof of Jaikin-Zapirain's theorem, valid for all primes p. In particular, this also settles a conjecture of Jaikin-Zapirain that the result holds for p = 2. The proof involves projective representations of finite groups as well as a rationality result from the model theory of the p-adic numbers. Such model theoretic rationality results have been proved and used by Hrushovski, Martin, Rideau and Cluckers to establish, among other things, rationality of twist representation zeta functions of nilpotent groups (counting representations up to one-dimensional twists). Our techniques extend to also prove virtual rationality of twist representation zeta functions of groups such as GL_n(Z_p).

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Michael Schein is inviting you to a scheduled Zoom meeting.

Topic: BIU Algebra Seminar -- Stasinski
Time: Dec 16, 2020 11:00 AM Jerusalem

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