Modularity of Calabi--Yau varieties: 2011 and beyond

TL;DR

This paper reviews the progress and current status of the modularity of Calabi-Yau varieties of dimension up to three, covering arithmetic, geometric, and moduli aspects since 2003.

Contribution

It provides a comprehensive update on various types of modularity related to Calabi-Yau varieties, integrating recent developments since 2003.

Findings

Advances in automorphy of Galois representations

Progress in modularity of Picard--Fuchs solutions

Development in modular generating functions

Abstract

This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard--Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and (4) the modularity of moduli for families of Calabi-Yau varieties.