## Final Report Summary - AAMOT (Arithmetic of automorphic motives)

Number theory is, among other things, the study of solutions to polynomial equations. Two vast research programs, both initiated nearly 50 years ago, propose frameworks for organizing this study. The Langlands program postulates that the structure of solutions to equations is modeled by the theory of automorphic forms, whose origins lie in the study of highly symmetric families of differential equations, and thus ultimately in mathematical physics. Grothendieck's hypothetical theory of motives proposes a framework based on geometry, specifically algebraic geometry. Automorphic motives are the structures that are encompassed both by the Langlands program and by Grothendieck's theory. All motives are conjecturally automorphic motives. This conjecture is out of reach at present and is likely to remain out of reach for decades. However, a large class of automorphic motives can be constructed by means of a close study of the geometry of Shimura varieties, which are highly symmetric geometric objects whose properties can be explained in terms of automorphic forms. Essentially all of the examples on which the conjectures are based have been provided by Shimura varieties, and they remain the preferred context in which the conjectures are tested and proved.

The project AAMOT is devoted to a systematic study of the automorphic motives arising from the geometry of Shimura varieties. The main goals are to extend the class of available examples to its natural limit and to apply the full range of analytic, geometric, and algebraic methods to verify outstanding conjectures for these motives. The most important result obtained thus far is the construction of the p-adic Galois realizations of motives that are expected to correspond under global Langlands reciprocity to cohomological automorphic representations of GL(n) over a CM field. This was obtained in collaboration with Kai-Wen Lan, Richard Taylor, and Jack Thorne, and has since been extended in a spectacular way by Peter Scholze. Other important results include the complete construction of p-adic L-functions for unitary groups, in collaboration with Ellen Eischen, Jian-Shu Li, and Chris Skinner, and the analysis of special values of L-functions of adjoint and tensor product motives, obtained in collaboration with Harald Grobner, Erez Lapid, and Lin Jie. All of these results were proved by applying techniques in the geometry of Shimura varieties. The scope of these results has been expanded by the collaborative project of Kaletha, Minguez, Shin, and White, to determine the endoscopic classification of automorphic representations of unitary groups.

The project with Böckle, Khare, and Thorne represents a new direction in the study of automorphic Galois representations, in the setting of Vincent Lafforgue's approach to the Langlands correspondence over function fields.

The PI's project initiated with Akshay Venkatesh in the Fall of 2014 in Berkeley has expanded into an extended collaboration with Henri Darmon and Victor Rotger; it indicates another direction for the development of the arithmetic of automorphic motives that brings together all the main themes of AAMOT.

An outstanding open question is the complete characterization of automorphic motives. This is the setting of the Fontaine-Mazur Conjecture, which states that certain kinds of motives of rank 2 automatically arise from elliptic modular forms, which are the simplest of all automorphic forms. The proof of generalizations of the Fontaine-Mazur conjecture is now a central goal of algebraic number theory. The known cases, due largely to Kisin and Emerton (as well as Khare and Wintenberger), are based on the p-adic local Langlands correspondence developed by Breuil, Colmez, Paskunas, and their collaborators. Progress on more general versions of the Fontaine-Mazur Conjecture seems to require a generalization of the p-adic local Langlands correspondence, and this in turn is most likely to proceed by means of geometrization, along the lines of the geometric Langlands program. The final goal of AAMOT is to find an appropriate framework for this geometrization. Although no definitive results have yet been proved, AAMOT has contributed to the advance of the geometrization program by the PI's collaborative efforts with a number of colleagues, through its support for the Fall 2014 program on New Geometric Methods in Number Theory and Automorphic Forms at the Mathematical Sciences Research Institute in Berkeley, which ran concurrently with a program on Geometric Representation Theory, and through its organization of a workshop at the CRM "Ennio de Giorgi" in Pisa in June 2016.

The project AAMOT is devoted to a systematic study of the automorphic motives arising from the geometry of Shimura varieties. The main goals are to extend the class of available examples to its natural limit and to apply the full range of analytic, geometric, and algebraic methods to verify outstanding conjectures for these motives. The most important result obtained thus far is the construction of the p-adic Galois realizations of motives that are expected to correspond under global Langlands reciprocity to cohomological automorphic representations of GL(n) over a CM field. This was obtained in collaboration with Kai-Wen Lan, Richard Taylor, and Jack Thorne, and has since been extended in a spectacular way by Peter Scholze. Other important results include the complete construction of p-adic L-functions for unitary groups, in collaboration with Ellen Eischen, Jian-Shu Li, and Chris Skinner, and the analysis of special values of L-functions of adjoint and tensor product motives, obtained in collaboration with Harald Grobner, Erez Lapid, and Lin Jie. All of these results were proved by applying techniques in the geometry of Shimura varieties. The scope of these results has been expanded by the collaborative project of Kaletha, Minguez, Shin, and White, to determine the endoscopic classification of automorphic representations of unitary groups.

The project with Böckle, Khare, and Thorne represents a new direction in the study of automorphic Galois representations, in the setting of Vincent Lafforgue's approach to the Langlands correspondence over function fields.

The PI's project initiated with Akshay Venkatesh in the Fall of 2014 in Berkeley has expanded into an extended collaboration with Henri Darmon and Victor Rotger; it indicates another direction for the development of the arithmetic of automorphic motives that brings together all the main themes of AAMOT.

An outstanding open question is the complete characterization of automorphic motives. This is the setting of the Fontaine-Mazur Conjecture, which states that certain kinds of motives of rank 2 automatically arise from elliptic modular forms, which are the simplest of all automorphic forms. The proof of generalizations of the Fontaine-Mazur conjecture is now a central goal of algebraic number theory. The known cases, due largely to Kisin and Emerton (as well as Khare and Wintenberger), are based on the p-adic local Langlands correspondence developed by Breuil, Colmez, Paskunas, and their collaborators. Progress on more general versions of the Fontaine-Mazur Conjecture seems to require a generalization of the p-adic local Langlands correspondence, and this in turn is most likely to proceed by means of geometrization, along the lines of the geometric Langlands program. The final goal of AAMOT is to find an appropriate framework for this geometrization. Although no definitive results have yet been proved, AAMOT has contributed to the advance of the geometrization program by the PI's collaborative efforts with a number of colleagues, through its support for the Fall 2014 program on New Geometric Methods in Number Theory and Automorphic Forms at the Mathematical Sciences Research Institute in Berkeley, which ran concurrently with a program on Geometric Representation Theory, and through its organization of a workshop at the CRM "Ennio de Giorgi" in Pisa in June 2016.