Mathematics Colloquium

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Bias in Cubic Gauss Sums

Speaker: Maksym Radziwill, Northwestern University

Location: Warren Weaver Hall 1302

Date: Monday, April 14, 2025, 3:45 p.m.

Synopsis:

Quadratic Gauss sums are defined as \sum_{x mod q} \exp(2\pi i x^2 / q). One can view these as finite field analogues of Gaussian integrals, besides they are quite useful, for example one can derive quadratic reciprocity from properties of Quadratic Gauss sums. Kummer, motivated by attempts at deriving cubic reciprocity, was interested in the properties of Cubic Gauss sums, defined as \sum_{x mod q} \exp(2 \pi i x^3 / q). Kummer noticed a numerical bias in the signs of these (real-valued) sums. The existence of this bias perplexed number theorists for a long time. Patterson was the first to realize that cubic Gauss sums can be realized as coefficients of very exotic (weight 1/3) automorphic forms. This led to a conjectural explanation of the bias that Kummer observed, the existence of which was finally confirmed (conditionally on the Generalized Riemann Hypothesis) in recent-ish work of the speaker with Alex Dunn. I will discuss the circle of ideas surrounding cubic Gauss sums, and the reasons why they remain both interesting and mysterious.