Darmon Project

More details on the project will be provided here as they become available.

The working group will be led by Owen Barrett, Lea Beneish, and Thomas Browning. To participate in this working group, send an email to one of these organizers.

Both students and faculty are encouraged to participate in the working groups.

The project meets at 1:30-5 pm at Evans 730.

Title:

Rigid meromorphic cocycles

Abstract:

The goal of this lecture series is to describe a possible framework for extending the

classical theory of complex multiplication to the setting of real quadratic base fields.

The theory of complex multiplication asserts that the values of modular functions

(with rational fourier expansions, say) at imaginary quadratic arguments of the Poincaré

upper half plane belong to abelian extensions of the relevant quadratic imaginary field, and

indeed that these values generate all possible such extensions. This leads to invariants

that have found a variety of arithmetic applications, most notably:

— Singular moduli: the values of the j function at quadratic imaginary arguments. Their factorisations

were studied by Gross and Zagier, revealing rich patterns and an intriguing connection with certain modular

generating series;

— Elliptic units: the values of modular units at quadratic imaginary arguments, whose logarithms are related to

derivatives of L-series of abelian characters of imaginary quadratic fields, via the Kronecker limit formula. These

special units are the basis for a proof of essentially the only known cases of Stark’s conjecture, and also play a key role

in the work of Coates-Wiles and Rubin on the arithmetic of elliptic curves with complex multiplication;

— Heegner points: the values of modular parametrisations of elliptic curves at quadratic imaginary arguments,

whose heights are related to derivatives of Hasse-Weil L-series of elliptic curves, via the Gross-Zagier formula. These

special points are the basis for a proof of the (weak) Birch and Swinnerton Dyer conjecture for elliptic curves over

Q whose Hasse-Weil L-series admits at most a simple order zero at the center.

The approach to obtain analogous invariants for real quadratic base fields (the real quadratic singular moduli of

[DV], the Gross-Stark units attached to p-adic L-series of odd characters of a real quadratic fields, and Stark-Heegner

points, respectively) involves replacing complex analysis by p-adic (rigid) analysis, and modular functions by rigid meromorphic cocycles. These are simply one-cocycles on the Ihara group

SL(2, Z[1/p]) with values in the multiplicative group of rigid meromorphic functions on Drinfeld’s p-adic upper half-plane.

The lecture series will describe these mathematical objects and their role in the theory of ‘’real multiplication’’, with an emphasis on a concrete description that will allow us to compute with them in practice (on the computer).

References:

[DV]: Henri Darmon and Jan Vonk.

Singular moduli for real quadratic fields: a rigid analytic approach.

Duke Math Journal, 170, Number 1 (2021), 23-93.

Goal for working group:

In this article, a lot of numerical examples of rigid meromorphic cocycles and their values are computed,

when the prime p is 2,3,5,7, or 13, (or, possibly, when p divides the cardinality of the Monster group). The reason for this goes back to Andrew Ogg’s observation that these are precisely the primes for which the modular curve X_0(p)

(or its quotient by the Atkin-Lehner involution) are of genus zero. For other primes, the modular forms

of weight two and level p give rise to obstructions for producing rigid meromorphic cocycles "with prescribed divisor’’.

Indeed, there is an analogue of Borcherds' theory of singular theta lifts, which assigns rigid meromorphic

cocycles to the principal parts of weakly holomorphic modular forms of weight 1/2 and level 4p.

The goal of the working group project will be to compute rigid meromorphic cocycles for other primes p

for which the space of weight two cusp forms of level p is non-trivial, with the goal of illustrating the ``real quadratic

Borcherds theory” with explicit examples, and extending the scope of the numerical experiments

that were performed in [DV].

Suggested preparation for the working group:

Browse through DV and the references therein to get a general impression of the topics that will be covered

Gain some familiarity using Pari or Sage to do number theoretic computations. (Another alternative is Magma.)