Number Theory Seminar @ University of Melbourne

The theory of newforms, due to Atkin and Lehner, provides a powerful method of decomposing spaces of modular forms. However, many problems occur when trying to generalise this theory to characteristic p. Recently, Deo and Medvedovsky have suggested a way around these problems by using purely algebraic notions to define newforms. In this talk, we summarise the results of Deo and Medvedovsky and discuss possible generalisations of their work.

This is a talk on basic definition of p-adic L-functions of Kubota and Leopoldt.

I’ll begin with Bost’s generalization of the p-curvature conjecture, and describe the classical geometric concepts at play such as connections, parallel transport, and associated horizontal subbundle. This naturally motivates the discovery of the Gauss-Manin connection on algebraic de Rham cohomology, and its non-abelian analogue due to Simpson. I’ll state Katz’s theorem that the p-curvature conjecture (equivalently, Bost's conjecture) holds for the abelian Gauss-Manin connection, and outline the ingredients to make that statement, such as the Hodge filtration, conjugate filtration, and Kodaira-Spencer map. I’ll then define non-abelian analogues of these objects, and state a theorem which suitably equates them. This is the non-abelian analogue of part 1 (the characteristic p step) of Katz’s theorem, and is progress towards proving Bost for the non-abelian Gauss-Manin connection. There is also a transcendental characteristic 0 part 2 of Katz' theorem, but its non-abelian analogue is more mysterious.

The absolute Galois group of a local or global field can be better understood by studying its representations, important classes of which are constructed from geometric objects such as elliptic or modular curves. The results of a line of work initiated by Serre, Momose and Ribet suggest that certain interesting symmetries of a Galois representation constructed this way are in bijection with the symmetries of its underlying geometric object. We present a recent result in this direction for two-dimensional representations, obtained in a joint work with J. Lang and A. Medvedovsky. We also hint at what is known and expected in the higher-dimensional case, and how this can be interpreted in terms of p-adic Langlands functoriality.