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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - GMLP (Global Methods in the Langlands Program)

Teaser

Summary

In May 2018, mathematician Robert Langlands travelled to Norway in order to receive the Abel Prize from King Harald V. The prize was awarded to recognize Langlands\' creation of and contributions to what is nowadays called the Langlands Program, going back to the 1960\'s. In its most basic form, the Langlands form posits a duality in number theory: a duality between automorphic forms and motives, which are linked by their interaction with prime numbers. Many of the greatest achievements of number theory in the last 400 years can be seen through the lens of the Langlands program, starting with Gauss\' quadratic reciprocity law, up to Wiles\' proof of Fermat\'s Last Theorem, and beyond. Most aspects of the Langlands Program remain conjectural, and its study is a central topic in modern number theory.

This project has its roots in recent work of Vincent Lafforgue, which gives an entirely new perspective on the Langlands Program. Lafforgue used algebraic geometry to describe a whole new set of symmetries which act on automorphic forms, which he calls excursion operators. Using these symmetries, he was able to give a very elegant proof of a large part of Langlands\' conjectures, in the context of algebraic function fields. The overall goal of this project is to explore the new avenues of investigation made possible by Lafforgue\'s striking discoveries. A fundamental question that motivates much of our work is: what part of the number theoretic world plays a role in Lafforgue\'s new Langlands correspondence? Conjectures state that every part of this world should be visible using the theory of automorphic forms. We aim to show that this is the case.

Work performed

\"The main achievement of the project so far has been to prove a \"\"potential\"\" version of the Langlands correspondence in many cases. Lafforgue\'s construction associates to any automorphic representation (an object with many symmetries of a number theoretic nature) over an algebraic function field a Galois representation (a realization of the symmetries satisfied by numbers themselves). The dream would be to show that any Galois representation arises this way. Our \"\"potential\"\" theorem shows that this is indeed the case, after passage to a finite index subgroup of the Galois group.

Another key achievement, of a quite different nature, has been a study of the arithmetic statistics of abelian surfaces. These are a 2-dimensional analogue of the 1-dimensional elliptic curves, which are among some of the most famous objects in diophantine (or arithmetic) geometry. We prove a result about the average size of the 3-Selmer group of an abelian surface. The definition of the Selmer group is technical, but our results have the concrete consequence that a positive density of monic degree 5 polynomials with rational coefficients take no rational square values. The technical heart of our work uses ideas from algebraic geometry that have recently played an important role in the Langlands program (in particular, in the work of Ngo on the fundamental lemma and in Lafforgue\'s work).

The third main result concerns the Langlands Program over number fields, which is of more direct interest for studying diophantine equations. As part of a large (10 author) collaboration, we have proved the first potential modularity theorems for elliptic curves, and their symmetric power Galois representations, over imaginary quadratic fields. This means that many results such as the Sato--Tate conjecture, formerly known only for elliptic curves over totally real fields (such as the rational numbers), can be shown to hold in this context.
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Final results

\"We have set some ambitious goals for the remainder of the project period. The first will be to implement a strategy to use our potential automorphy theorems for Lafforgue\'s global correspondence to prove results about the local Langlands correspondence constructed by Genestier--Lafforgue. We expect this to have applications both to local fields of positive characteristic and, in some cases, to local fields of characteristic zero (such as the p-adic rationals).

Another will be to go further in our study of the modularity of elliptic curves over imaginary quadratic fields. Now that they have been shown to be potentially modular, it is natural to ask if one can prove they are modular, and perhaps to look for applications such as the analogue of Fermat\'s Last Theorem over imaginary quadratic fields. There is much still to be done here.

An exciting new direction will be the study of new cases of Langlands\' functoriality conjectures in the context of number fields. The most basic example, that of symmetric power functoriality for GL(2), has been studied in depth in the \"\"potential\"\" context -- indeed, this is the real content behind theorems like the Sato--Tate conjecture for elliptic curves. Proving functoriality over the base field (without the \"\"potential\"\" modifier) is much harder, but carries its own rewards (for example, the analytic, as opposed to merely meromorphic, continuation of the L-functions which appear in the Birch--Swinnerton-Dyer conjecture). We will implement new ideas about the p-adic geometry of so-called eigenvarieties to prove infinitely many new cases of Langlands functoriality.\"