Title：An Approximate Form of Artin's Holomorphy Conjecture and Nonvanishing of Artin $L$-functions
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：Prof.J esse Thorner （University of Illinois, Urbana-Champaign）
Time：10:30-11:30 am, March 11, 2021
Place: Zoom, Meeting ID: 881 9278 2209, Passcode: 705185
