分野別セミナー

[講演概要]
Artin L-functions are associated to irreducible non-trivial characters of the Galois group of a normal extension of
number fields. Conjecturally, Artin L-functions are holomorphic and non-vanishing except on the critical line.
Together with Robert Lemke Oliver and Jesse Thorner, we unconditionally prove for many families of Artin
L-functions, all except few of them are holomorphic and non-vanishing in a wide region. The number of exceptions is quantified in terms of a field counting problem which, in many cases of interest, is provably small. This has
applications to extremal class numbers, counting prime ideals in degree n S_n-extensions, and the subconvexity
problem for Dedekind zeta functions. I will outline the main result, some applications, and key obstacles in the proof.