Real quadratic singular moduli and $p$-adic families of modular forms

TL;DR

This paper explores the analogy between complex multiplication for imaginary quadratic fields and a conjectural real multiplication theory for real quadratic fields, focusing on $p$-adic methods and modular forms.

Contribution

It surveys the theory of complex multiplication and discusses conjectural real quadratic analogues involving rigid meromorphic cocycles and $p$-adic families of modular forms.

Findings

Rigid meromorphic cocycles relate to real quadratic points as analogues of singular moduli.

Comparison of complex multiplication theory with conjectural real quadratic counterparts.

Emphasis on the role of modular form families in both classical and conjectural settings.

Abstract

The classical theory of elliptic curves with complex multiplication is a fundamental tool for studying the arithmetic of abelian extensions of imaginary quadratic fields. While no direct analogue is available for real quadratic fields, a (conjectural) theory of "real multiplication" was recently proposed by Darmon and Vonk, relying on $p$-adic methods, and in particular on the new notion of rigid meromorphic cocycles. A rigid meromorphic cocycle is a class in the first cohomology of the group $\mathrm{SL}_2(\mathbb{Z}[1/p])$ acting on the non-zero rigid meromorphic functions on the Drinfeld $p$-adic upper half plane by M\"obius transformation. The values of rigid meromorphic cocycles at real quadratic points can be thought of as analogues of singular moduli for real quadratic fields. In this survey article, we will discuss aspects of the theory of complex multiplication and compare them with conjectural analogues for real quadratic fields, with an emphasis on the role played by families of modular forms in both settings.