Modular forms, projective structures, and the four squares theorem

TL;DR

This paper presents a geometric approach to Lagrange's four-square theorem using modular forms and projective differential geometry, offering a novel proof that avoids complex analysis.

Contribution

It introduces a new geometric proof of Lagrange's four-square theorem based on projective differential geometry, diverging from traditional analytical methods.

Findings

New geometric proof of four-square theorem

Connection between modular forms and projective geometry

Simplification of proof avoiding complex analysis

Abstract

It is well-known that Lagrange's four-square theorem, stating that every natural number may be written as the sum of four squares, may be proved using methods from the classical theory of modular forms and theta functions. We revisit this proof. In doing so, we concentrate on geometry and thereby avoid some of the tricky analysis that is often encountered. Guided by projective differential geometry we find a new route to Lagrange's theorem.