### 26 May, 2020

Defining Newforms in Characteristic p

Daniel Robert Johnston (the 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.

### 12 May, 2020

p-adic L-functions

Chenyan Wu (the University of Melbourne)

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

### 17 Mar., 2020 [Cancelled!]

On Mœglin’s Parametrization of Arthur Packets for p-adic Quasisplit Sp(N) and SO(N) following Bin Xu's paper

Chenyan Wu (the University of Melbourne)

### 10 Mar., 2020

Bost’s p-curvature conjecture for the Gauss-Manin connection on
non-abelian de Rham cohomology

Max Menzies (Tsinghua University, Beijing, China)

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.

### 17 Feb., 2020

Special hour: 10:00 - 11:00

Images and symmetries of Galois representations

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.