Quadratic residue patterns, algebraic curves, and a K3 surface (to the memory of Lydia Goncharova)

Joint work with V. Kirichenko, S. Vladuts, and I. Zacharevich

Quadratic residue (modulo a prime) patterns are studied from the end of 19^th century. My talk consists of two formally independent parts, closely related in spirit.

The first part is devoted to the classical problem on strings of consecutive quadratic residues. We reduce this problem to counting points on elliptic and hyperelliptic curves, thus getting results unavailable by classical methods.

In the second part I shall state the last unpublished result of Lydia Goncharova on the sets of residues such that the difference between any two elements is a quadratic residue. We did not succeed trying to restore her elementary proof, but we managed to prove her theorem reducing it to the problem of counting points on a very specific K3 surface.