# Manchester Algebra Seminar - Alexander Stasinski

Starts: | 13:00 23 Nov 2021 |
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Ends: | 14:00 23 Nov 2021 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

Speaker: Alexander Stasinski - Durham

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

Abstract: 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 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. Our techniques extend to also prove virtual rationality of twist representation zeta functions of groups such as GL_n(Z_p).

Place: Frank Adams (and to be streamed online*)

- subject to equipment and connection

Tea and biscuits 12:45 in the foyer