Since it was initiated by the famous Indian mathematician Ramanujan more than a century ago, the arithmetic theory of modular forms has become a major field of mathematical research that plays a central and unifying role in our modern understanding of number theory and algebraic geometry (motives) and has connections with various fields of mathematics. The simplest and oldest reason for which modular forms are useful to those other fields is that many sequences of integers of interest in number theory, combinatorics, algebra and representation theory, algebraic geometry, and theoretical physics have the surprising property that their generating function satisfies a functional equation that makes it a modular form. Hence the study of the arithmetic properties of these sequences becomes a part of the study of arithmetic property of modular forms. In particular, to study divisibility or congruence properties of those sequences modulo a prime number p, one must study the theory of modular forms modulo p.
The project proposes a new approach to this fundamental study, based on the determination of the structure of the big Hecke algebras acting on space of modular forms modulo p, and the existence of a Galois pseudo-representation attached to every modular form, not necessarily an eigenvector for the Hecke operators. This approach should allow for a systematic theory of congruences between the coefficients of weakly holomorphic modular forms, not only producing, as was often the case isolated examples of congruences, but leading to a complete classification of such congruences. It will also allow to completely describe the asymptotic behavior of coefficients of holomorphic modular forms modulo p, which is only known in certain very particular cases as of now, and, in the more mysterious case of weakly holomorphic modular forms, to better understand the "chaos" that conjecturally lurks behind these congruences, in particular leading to the proof of several outstanding conjectures concerning the reduction modulo primes p of the partition function. A second aspect of the project, which is distinct from the first but uses similar tools, aims at proving important cases of the Bloch-Kato conjecture relating special values of L-functions of motives and Selmer groups, and at developing the theory of p-adic L-functions. In the long run, those two aspects should be reunited into one very large theory of universal families of automorphic forms in mixed characteristic, including at the same time the Galois, L-functions, and coefficients aspects.
