## London Heilbronn Colloquium 2023: Henri Darmon

13 Nov 2023, by Events in15 December 2023 at 16:00 (followed by wine reception)

**Venue**: Anatomy Lecture Theatre, Level 6 of the main King’s Building, King’s College London, Strand, London, WC2R 2LS

[Best accessed from the Strand at the entrance just south of St Mary Le Strand Church], see map here

**Organiser**: Alex Torzewski (KCL)

The London Heilbronn Colloquia are a series of triannunal lectures by distinguished mathematicians and theoretical physicists at the forefront of current research. They are aimed at a general mathematical audience and informal interaction with the speaker.

**Explicit Class Field Theory**

Henri Darmon, Distinguished James McGill Professor, Mathematics and Statistics, McGill University, Canada

Two of the most striking discoveries of 18th and 19th century number theory are the Kronecker-Weber theorem and the theory of complex multiplication. The first asserts that the maximal abelian extension of the field Q of rational numbers is generated by roots of unity – in other words, that all abelian extensions of Q can be constructed by adjoining values at rational arguments of the transcendental function

e(z) := e^{2pi iz} = cos(2 pi z) + i sin(2 pi z).

The second achieves something similar for a quadratic imaginary field K, constructing essentially all of its abelian extensions from values of the modular j-function at arguments of K.

Finding analytic functions that would play the role of trigonometric and modular functions in generating abelian extensions, or class fields, of more general base fields is the somewhat loosely formulated program of explicit class field theory, also known as Kronecker’s Jugendtraum or Hilbert’s twelfth problem.

Partial progress was achieved with the theory of complex multiplication of abelian varieties initiated by Hilbert and his school and brought to maturity in the eponymous 1961 treatise of Shimura and Taniyama. Through this theory, class fields of CM fields are obtained from the values of modular functions at points attached to the moduli of CM abelian varieties in suitable (Hilbert, Siegel, orthogonal, . . .) Shimura varieties.

Hilbert’s twelfth problem for non-CM base fields remains shrouded in a great deal of mystery, hinting—perhaps—at a rich function theory for arithmetic quotients even when the underlying real symmetric space fails to be endowed with a complex structure and hence cannot uniformise a Shimura variety. This may be what Hilbert intuited when he declared, in his celebrated 1900 address at the Paris ICM,

“I am certain that the theory of analytical functions of several variables in particular would be notably enriched if one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field.”

This talk will describe some possible substitutes for trigonometric and modular functions, with applications to explicit class field theory beyond the traditional framework of CM fields.

**About the Speaker:** Henri Darmon is a number theorist who works on Hilbert’s 12th problem[1] and its relation with the Birch–Swinnerton-Dyer conjecture. He is currently a James McGill Professor of Mathematics at McGill University. Darmon was elected to the Royal Society of Canada in 2003. In 2008, he was awarded the Royal Society of Canada’s John L. Synge Award. He received the 2017 AMS Cole Prize in Number Theory for his contributions to the arithmetic of elliptic curves and modular forms, and the 2017 CRM-Fields-PIMS Prize, which is awarded in recognition of exceptional research achievement in the mathematical sciences.

Information on past and future colloquia is available here.

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