# Number Theory Seminar @ University of Melbourne

## 14:00 - 15:00 Wednesdays

Alex Ghitza and I are running a working seminar on things number-theoretic.
Unless otherwise specified, we meet at 14:00 - 15:00 Wednesdays in Semester 2, 2020. Zoom links are announced via mailing list. The current focus is to study the Jacquet-Langlands Theory. This semester's seminar has been concluded. We will resume in in Semester 1, 2021.

You can be informed of future talks by subscribing to the mailing list (enter your preferred email address in the textbox labeled "Your email address", choose the action "Subscribe" below it and click the button "Go!").

For the record of life before Covid-19, we have been meeting at 11am Tuesdays in Semester 1, 2020, in room 107 (Peter Hall building).

### 9 Dec., 2020

Local L-factors for GL(2)

Chenyan Wu (the University of Melbourne)

### 2 Dec., 2020

Local gamma factors and review of L-functions for GL(1).

### 28 Oct., 2020

Functional Equation of Zeta Integrals

### 21 Oct., 2020

Even More on Principal Series Representations

Chenyan Wu (the University of Melbourne)

### 14 Oct., 2020

More on Principal Series Representations

Caveat: Melbourne has entered Summer time. Double check your timezone conversion.

Notes
Chenyan Wu (the University of Melbourne)

### 30 Sept., 2020

Criterion and examples of Supercuspidal Representations

Following Bushnell and Henniart, The Local Langlands Conjecture for GL (2).

Notes,

An example
Finn McGlade (the University of Melbourne)

### 23 Sept., 2020

Principal Series Representations and Supercuspidal Representations

Notes: see updated version for 30 Sept.

Finn McGlade (the University of Melbourne)

### 16 Sept., 2020

Contragredient Representations

Milton Lin (the University of Melbourne)

### 9 Sept., 2020

Whittaker Model and Jacquet Functor

Milton Lin (the University of Melbourne)

### 2 Sept., 2020

Kirillov Model, Part II

Alex Ghitza (the University of Melbourne)

### 26 Aug., 2020

Kirillov Model, Part I

Alex Ghitza (the University of Melbourne)

### 19 Aug., 2020

Introduction to Admissible Representations of GL(2) and Kirillov Model

Chenyan Wu (the University of Melbourne)

### 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.