Modular Arrangements

TL;DR

This paper explores the arithmetic properties of modular curves, particularly their periods of mixed Hodge structures, using the square of the modular curve as a key object, linking geometric, number-theoretic, and analytic perspectives.

Contribution

It investigates the arithmetic properties of the square of modular curves and their associated mixed Hodge structures, connecting geometric definitions with number-theoretic analysis.

Findings

Analysis of periods of mixed Hodge structures on modular curve squares

Identification of the square as a moduli space of split abelian surfaces

Application of analytic techniques to arithmetic geometry problems

Abstract

The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the perspective of number theory. Furthermore, there is a well-developed and powerful analytic technique available. We will use the square of the modular curve as the experimental object to investigate the arithmetic properties of the periods of mixed Hodge structures. There is an additional reason for this study: this square is naturally associated with a family of (Hecke) curves. These curves form components of the Neron-Severi locus, allowing for the interpretation of the square of the moduli curve as the moduli space of split (i.e., the product of two elliptic curves) abelian surfaces.