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Frank Thorne : Secondary Terms in Counting Functions for Cubic Fields

I will speak about recent progress on the enumeration of number fields, with particular attention to joint work with Taniguchi, which proved the existence of a negative secondary term in the counting function for cubic fields by discriminant. Among other results, we also found surprising biases in arithmetic progressions -- e.g., cubic field discriminants are more likely to be 5 (mod 7) than 3 (mod 7). Our work applies the analytic theory of the Shintani zeta function, which I will describe briefly. I will also discuss other approaches to related questions (and in particular an independent, and different, proof of the secondary term due to Bhargava, Shankar, and Tsimerman), using approaches as diverse as the geometry of numbers, algebraic geometry, and class field theory.

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