Singular moduli for real quadratic fields

Abstract: In the early 20th century, Hecke studied the diagonal restrictions of Eisenstein series over real quadratic fields. An infamous sign error caused him to miss an important feature, which later lead to highly influential developments in the theory of complex multiplication (CM) initiated by Gross and Zagier in their famous work on Heegner points on elliptic curves. In this talk, we will explore what happens when we replace the imaginary quadratic fields in CM theory with real quadratic fields, and propose a framework for a tentative 'RM theory', based on the notion of rigid meromorphic cocycles, introduced in joint work with Henri Darmon. I will discuss several of their arithmetic properties, and their apparent relevance in the study of explicit class field theory of real quadratic fields, the construction of rational points on elliptic curves, and the theory of Borcherds lifts. This concerns various joint works, with Henri Darmon, Alice Pozzi, and Yingkun Li.

algebraic geometrynumber theory

Audience: researchers in the topic

( slides )

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MIT number theory seminar

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