Reduction of plane quartics and Cayley octads

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(Q). For hyperelliptic curves y² = 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 P³ associated to the curve.