Talk by Matías Martres: From Sums of Squares to Theta Functions - July 16, 2026

On Thursday, July 16, 2026, Lic. Matías Martres gave a talk as part of the ‘4th Mathematics Students Meeting’.

Location and time: Room 101 at the Faculty of Engineering, UdelaR. 9:00 a.m.

Title: From Sums of Squares to Theta Functions

Abstract: Characterizing which integers can be represented as a sum of two squares is a classical problem in number theory. We will begin by studying this problem and showing that this characterization reduces to proving that: “if p is a prime of the form 4k+1, then it can be represented as a sum of two squares”. To this end, we will introduce the theta function associated with this problem, show that it is a modular form and see how this allows us to conclude.

Once the classical problem is solved, we will change our perspective. The same construction allows us to associate a theta function with any positive definite quadratic form, whose modularity is a theorem of Hecke and Schoeneberg. This generalization already suggests that we could try to study these functions in a broader framework