John Yin : Chebotarev Density Theorem for Local Fields

Posted by Zhehao Li , part of the Graduate Number Theory Seminar.

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April 19, 2023, 4 p.m.

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Abstract

For a given p, let rho(p) denote the proportion of irreducible quadratic polynomials over Z which generates field extensions which are totally split at p. Then, it turns out rho(t)=(t^2+1)/(2(t+1)^2). There are two remarkable things about this: it is a rational function in Q(t) and it satisfies the functional equation rho(t)=rho(-t). Bhargava, Cremona, Gajovic, and Fisher conjecture that the same holds for arbitrary degrees and splitting types. We prove a vast generalization of this conjecture in the tame case. This is joint work with Yifan Wei and Asvin G.

Zoom link: https://uic.zoom.us/j/82357749325?pwd=L1JoRm9rUjZJbDNZeUJvdDljWGdhUT09