Reduction of plane quartics and Cayley octads

Abstract: For a long time, number theorists have been interested in studying the reduction modulo $p$ of algebraic varieties defined over number fields. For example, in the case of an elliptic curve $E$, where we distinguish between good, multiplicative, and additive reduction, the Birch and Swinnerton-Dyer conjecture predicts that the reduction plays a crucial role in understanding the rank of $E(\mathbb{Q})$. For hyperelliptic curves $y^2 = f(x)$, the reduction has been studied extensively through the Weierstrass points, i.e. the roots of $f(x)$. In this talk, I will tell about recent work joint with Jordan Docking, Vladimir Dokchitser, Reynald Lercier, Elisa Lorenzo Garcia, and Andreas Pieper, in which we study the situation for the first case of non-hyperelliptic curves: plane quartics. As a result of numerous computations, we made a prediction how the reduction type of a plane quartic can be determined from the Cayley octad, a set of eight points in $\mathbb{P}^3$ associated to the curve.

number theory

Audience: researchers in the topic

UNSW Number Theory Seminar

Series comments: The UNSW Number Theory Seminar has been running in the School of Mathematics and Statistics, UNSW Sydney, since 2015. The seminar usually takes place every two weeks on Wednesdays, 2 - 4 pm, with two talks scheduled each seminar, followed by refreshments and discussions.