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

Quadratic residue (modulo a prime) patterns have been studied since the end of the 19th 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 obtaining 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In the second part, I shall state the last unpublished result of Lydia Goncharova on the sets of residues such that the difference between any two elements is a quadratic residue.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 in restoring her elementary proof, but we managed to prove her theorem by reducing it to the problem of counting points on a very specific K3 surface.
(Joint work with V. Kirichenko, S. Vladuts, and I. Zacharevich.)
 

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062