Modular functions and explicit class field theory: private reminiscences and public confessions

Abstract: The problem of constructing class fields of number fields from explicit values of modular functions has its roots in the theory of cyclotomic fields and the theory of complex multiplication. The latter theory acquired a renewed currency in the second half of the 20th century through its connections to the arithmetic of elliptic curves, manifested in the work of Coates--Wiles, Rubin, Gross--Zagier, and Kolyvagin.

I will give a personal account of my path towards a (slightly) better understanding of explicit class field theory for real quadratic fields and its applications to elliptic curves, taking advantage of the CHAT format to focus on the misconceptions, false starts, and dead ends that have marked my roundabout and tortuous, but also very enjoyable, mathematical journey so far.

number theory

Audience: researchers in the discipline

( video )

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CHAT (Career, History And Thoughts) series

Series comments: The CHAT series invite established professors to talk about either (1) their math career in general (2) their theorems or theories, but explained from a personal and historical perspective, like how they came up with the problem, what the Aha! moment was like, how the problem changes from its initial form to the published rigorous form.

The idea is that instead of talking about their latest theorems, the speakers would take a step back and talk about the trajectory of an idea, the path to the discovery of a theorem, the influence of ideas learned through a paper or a chance conversation with a colleague, and the hazards met and overcome along the way.

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