Recent Progress on Hilbert’s 12th Problem

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Recent Progress on Hilbert’s 12th Problem

 24 - 28 Jun 2024

ICMS, Bayes Centre, Edinburgh

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Deadline for public applications are now closed, no late applicants will be accepted

Scientific organisers

  • Adebisi Agboola, University of California Santa Barbara
  • Henri Darmon, McGill University
  • Benedict Gross, Harvard University
  • Alice Pozzi, Imperial College London

Keynote speakers

  • Elias Caeiro, École Normale Supérieure Paris
  • Pierre Charollois, Sorbonne Université
  • Michael Daas, University of Leiden
  • Samit Dasgupta , Duke University
  • Ellen Eischen, University of Oregon
  • Luis Garcia, University College London
  • Catherine Hsu, Swarthmore College
  • Mahesh Kakde, IISc Bangalore
  • Yukako Kezuka, Jussieu
  • Gene Kopp, Louisiana State Universtiy
  • Michael Lipnowski, Ohio State University
  • Pierre Morain, Sorbonne Université
  • Owen Patashnick, King's College London
  • Cristian Popescu, University of California, San Diego
  • Martí Roset Julià, McGill University
  • Takamichi Sano, Osaka Metropolitan University
  • Jan Vonk, University of Leiden
  • Jiyua Wang, University of Georgia
  • Robin Zhang, MIT

About:

 

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Hilbert’s twelfth problem asks for explicit constructions of the abelian extensions of a given number field, similar to what is known for the rational numbers and for imaginary quadratic fields. These abelian extensions are known as class fields because their Galois groups are identified with certain generalized ideal class groups. In the two known cases, the class fields are obtained via the adjunction of roots of unity and of torsion points on elliptic curves with complex multiplication. These are special values of complex analytic functions – the exponential function and elliptic functions with complex multiplication. Hilbert may have envisioned the use of special values of complex analytic functions to construct class fields of more general base fields.

In the 1970s, Harold Stark proposed a strikingly original approach to the generation of class fields, based on his conjectures on the leading term of Artin L-functions at s = 0. In the case of abelian L-functions with a simple zero at s = 0, Stark predicted that the first derivative was the logarithm of a unit in the respective class field, so exponentiating this derivative would give a generator for the abelian extension. In the two known cases, this reduced to the theory of circular and elliptic units, thanks to Dirichlet’s analytic class number formula and Kronecker’s limit formula. Although there is now extensive computational evidence that Stark’s conjecture is correct, there has been little progress on its solution. 

In the 1980s Benedict Gross formulated some p-adic  and tame  analogues of Stark’s conjectures, which gave more information on the p-adic expansions of the conjectural units. Since the p adic L-functions involved in Gross’s conjecture are related to certain Galois modules via the main conjecture in Iwasawa theory, these conjectures have proved more amenable than their complex analogs. Refinements of the Gross-Stark conjecture were later proposed by Darmon and Dasgupta, and the p-adic Gross -Stark conjectures were proved by Darmon, Dasgupta, Pollack and Ventullo around 2011. This line of argument has culminated in the recent work of Samit Dasgupta and Mahesh Kakde which, by proving a large part of the tame conjectures of Gross (along with the refinement of Darmon and Dasgupta in the broader setting of totally real fields) leads to a p−adic solution to Hilbert’s twelfth problem for this large class of fields. 

The goal of this workshop is to take stock of the recent work in this direction and of other progress around the theme of related approaches to explicit class field theory. The key to much of the progress over the years is the careful study of p-adic and tame deformations of modular forms, most notably, of Hilbert modular Eisenstein series. The p-adic interpolation of classical Eisenstein series was introduced by Jean-Pierre Serre to study the congruences of special values of L-functions and the construction of p-adic L-functions for totally real fields, and was further developed by Barry Mazur and Andrew Wiles in their proof of the main conjecture of Iwasawa theory. The workshop will focus on the breakthroughs of Dasguota and Kakde, with a lecture series by the two authors forming a cornerstone of the activity.

Some Recomended Reading
https://sites.math.duke.edu/~dasgupta/papers/Gross.pdf - A historical overview of the Gross-Stark conjecture. 

https://sites.math.duke.edu/~dasgupta/papers/ICM.pdf - A more detailed account of a recent work on the Brumer-Stark conjectures. 

  • Public Lecture, Thurs 27 June in The Bayes Centre: Prof Cristian D. Popescu, University of California, San Diego (details to follow) 

 

Programme

MONDAY 24 JUNE 2024
Registration and refreshments
Welcome and housekeeping
Samit Dasgupta, Duke University Introduction to the Brumer—Stark conjecture
Tea & coffee break
Catherine Hsu, Swarthmore College Explicit R=T via rank bounds
Tea & coffee break
Michael Lipnowski, Ohio State University Rigid meromorphic cocycles for orthogonal groups
Lunch
Luis Garcia, University College London The elliptic gamma function and units in number fields
Tea & coffee break
Pierre Charollois, Sorbonne Université On Eisenstein’s Jugendtraum for Complex Cubic Fields
Welcome Reception
TUESDAY 25 JUNE 2024
Mahesh Kakde , IISc Bangalore Proof of the Brumer—Stark conjecture away from p=2.
Tea & coffee break
Pierre Morain, Sorbonne Université Elliptic units above fields with exactly one complex place
Elias Caeiro, École Normale Supérieure Paris Stark-Heegner points and the class number one problem for families of real quadratic fields
Tea & coffee break
Owen Patashnick, King's College London Algebraic cycles and motives, a Jacobian romance
Lunch
Robin Zhang, MIT p-adic Shimura classes and Stark units
Tea & coffee break
Gene Kopp, Louisiana State Universtiy The Shintani–Faddeev modular cocycle: Stark units from q-Pochhammer ratios
Workshop dinner
WEDNESDAY 26 JUNE 2024
Mahesh Kakde, IISc Bangalore Proof of the Brumer—Stark conjecture away from p=2.
Tea & coffee break
Yukako Kezuka, Jussieu Non-commutative iwasawa theory of abelian varieties
Tea & coffee break
Jan Vonk, University of Leiden Rational points and RM invariants
Lunch and free afternoon
THURSDAY 27 JUNE 2024
Samit Dasgupta, Duke University The Brumer—Stark conjecture at p=2.
Tea & coffee break
Michael Daas, University of Leiden A p-adic analogue of a formula by Gross and Zagier
Martí Roset Julià, McGill University The Gross--Kohnen--Zagier theorem via $p$-adic uniformization
Tea & coffee break
Takamichi Sano, Osaka Metropolitan University On descent theory for Selmer complexes and applications to $p$-adic Birch and Swinnerton-Dyer conjectures
Lunch
Jiyua Wang, University of Georgia The residually indistinguishable case of Ribet's method for GL_2
Tea & coffee break
Samit Dasgupta, Duke University The Equivariant Tamagawa Number Conjecture
Public Lecture - Cristian D. Popescu, University of California, San Diego Number Theory: a brief history with a view towards Hilbert's 12th Problem
FRIDAY 28 JUNE 2024
Ellen Eischen, University of Oregon Beyond Hilbert modular forms and Hilbert’s Twelfth Problem
Tea & coffee break
Dominik Bullach, University College London On the refined `Birch--Swinnerton-Dyer type’ conjectures of Mazur and Tate
Tea & coffee break
Cristian Popescu, UC San Diego An Equivariant Main Conjecture in Iwasawa Theory