# 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.

Please go to the Number Theory Research Group website where you can find a link to join our mailing list. You will receive the Zoom link and announcements of talks.

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.