Description
Abstract: (Joint with Robert Lemke Oliver and Asif Zaman) 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).
The link for the talk is https://zoom.us/j/94752830725, the password is the order of $\mathrm{SL}_2(\mathbb{F}_{97})$.