Periodic Reporting for period 4 - GMLP (Global Methods in the Langlands Program) Reporting period: 2021-07-01 to 2022-06-30 Summary of the context and overall objectives of the project 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 Program 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 this project 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 have pushed the boundaries of our current understanding of the Langlands Program, resolving 50-year old questions concerning the existence of basic `functorial' families of automorphic forms, showing Lafforgue's Langlands correspondence does see all of the number theoretic world in a certain weak (`potential') sense, and creating and exploiting a new formalism which does give unconditional results towards the existence of a `local' Langlands correspondence. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The first main achievement of the project 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.The second 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. In a separate work, we also prove modularity (as opposed to potential modularity) theorems for elliptic curves, showing for the first time that a positive proportion of elliptic curves over any imaginary quadratic field are modular.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 final and most significant result we have obtained is the existence of all symmetric power liftings for holomorphic modular forms. This can be seen as the most important special case of Langlands' functoriality conjectures, an important part of the Langlands Program. We achieve this by combining results in the theory of p-adic automorphic forms, in particular the theory of eigenvarieties, with results on Selmer groups that rely in an essential way on Lafforgue's notion of pseudocharacter. This breakthrough work has been recognised by a number of major prizes. Aside from the usual lectures in at academic seminar and conferences, the results of this project have been the subject of lectures at major conferences (including the ICM and the CDM series at Harvard), a video interview, a podcast, and articles in popular scientific magazines. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The project has now ended. Jack Thorne
