Keynote Speaker:Jesse Thorner

Abstract:

Let $p$ be a prime, and let $\mathscr{F}_p(Q)$ be the set of number fields $F$ with $[F:\mathbb{Q}]=p$ with absolute discriminant $D_F\leq Q$. Let $\zeta(s)$ be the Riemann zeta function, and for $F\in\mathscr{F}_p(Q)$, let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The Artin $L$-function $\zeta_F(s)/\zeta(s)$ is expected to be automorphic and satisfy GRH, but in general, it is not known to exhibit an analytic continuation past $\mathrm{Re}(s)=1$. I will describe new work which unconditionally shows that for all $\epsilon>0$ and all except $O_{p,\epsilon}(Q^{\epsilon})$ of the $F\in\mathscr{F}_p(Q)$, $\zeta_F(s)/\zeta(s)$ analytically continues to a region in the critical strip containing the box $[1-\epsilon/(20(p!)),1]\times [-D_F,D_F]$ and is nonvanishing in this region. This result is a special case of something more general. I will describe some applications to class groups (extremal size, $\ell$-torsion) and the distribution of periodic torus orbits (in the spirit of Einsiedler, Lindenstrauss, Michel, and Venkatesh). This talk is based on joint work with Robert Lemke Oliver and Asif Zaman.

Speaker Introduction:

Professor Jesse Thorner (University of Illinois, Urbana-Champaign)

Inviter:

Lyu GuangshiProfessorofSchool of Mathematics

Time:

10:30-11:30 on March 11 (Thursday)

Location:

Zoom Meeting, ID: 881 9278 2209, password:705185

Hosted by: School of Mathematics, Shandong University

SDU-ANU Joint Science College