# Number Theory Seminar @ University of Melbourne

Alex Ghitza and I are running a working seminar on things number-theoretic.

Time: 13:00-14:00 Mon. (Melbourne local time) for 2022S2.

Venue: Peter Hall Building 107.

Zoom: Please join the mailing list to receive the Zoom link.

Topic: Automorphic representations. We plan to follow Getz and Hahn's book: An Introduction to Automorphic Representations with a view toward Trace Formulae. We will start from Chapter 3.

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 past semester 2022S1:

Time: 12:00-13:00 Fri. (Melbourne local time).

Venue: Outdoors on the Peter Hall Building lawn. Weather not permitting, Peter Hall Building 107.

Topic: Basic Notions in Modular Forms and Automorphic Forms.

### 26 Sept., 2022

Semester Break

### 19 Sept., 2022

Archimedean Representation Theory

### 12 Sept., 2022

(đť”¤, K)-modules and Admissible Representations

### 5 Sept., 2022

Representations of Lie groups, with a view towards understanding the algebraic approach to the archimedean components of automorphic representations

### 29 Aug., 2022

Automorphic Representations III

### 22 Aug., 2022

On the unramified spherical automorphic spectrum

I will give an introduction to common work with Marcelo De Martino and Eric Opdam, where we determine (for the moment in the number field case for a split group and supported in the trivial representation of the maximal torus) the automorphic spherical spectrum following the method initiated by Langlands using contour shift. It is based on a residue distribution approach developed previously by Eric Opdam and Gerrit Heckman. For exceptional groups, computer assistance was needed.

### 15 Aug., 2022

Automorphic Representations II

### 8 Aug., 2022

Moduli stacks of Galois representations

Recent years have seen a confluence between ideas coming from each of the arithmetic and geometric sides of the Langlands program. One manifestation of this is the rising importance of the consideration of moduli stacks of local and global Galois representations and related objects, going under the general heading of Langlands parameters.

In this talk I will describe these moduli stacks --- their definitions and basic structure, illustrated through examples --- in various local and global contexts. I will also briefly indicate the role they play in contemporary formulations of the Langlands program.

The talk will explain the work of many different researchers, but will include discussions of recent and ongoing work of the speaker in collaboration with Toby Gee and Xinwen Zhu.

### 4 Aug., 2022

A multiplicity formula of `K`-types

Special joint seminar with Representation Theory Seminar. Note special date and time: 15:15 - 17:00, 4 Aug. Location is still 107 Peter Hall Building.

In this talk, by using the trace formula method, I will prove a multiplicity formula of `K`-types for all representations of real reductive groups in terms of the Harish-Chandra character.

### 1 Aug., 2022

Automorphic Representations I

### 17 June, 2022

A Gross-Zagier Formula for CM cycles over Shimura Curves

Note: Special location at Evan Williams G03. This is the last talk of 2022S1. We will resume in Semester 2.

This talk is based on my thesis work and joint work (in progress) with Congling Qiu. I will introduce a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of CM cycles in Kuga varieties over Shimura curves with the derivatives of the L-functions associated to weight-2k modular forms. As a key original ingredient of the proof, I will introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function, which we apply to compute suitable integral extensions of the CM cycles, and local intersections at primes where the quaternion algebra associated with the Shimura curve ramifies.

### 3 June, 2022

Automorphic Forms, a Generalisation of Modular Forms, Part 2

Chenyan Wu (the University of Melbourne)

### 27 May, 2022

Iwasawa Theory of Class Group Schemes

Note: Joint with the Pure Maths Seminar; special Time 11:30 - 13:00

Iwasawa theory is the study of the growth of arithmetic invariants in
Galois extensions of global fields with Galois group a `p`-adic Lie group. Beginning with Iwasawa's seminal work
in which he proved that the `p`-primary part of the class group in â„¤_{p}-extensions of
number fields grows with striking and unexpected regularity, Iwasawa theory has become a central
strand of modern number theory and arithmetic geometry. While the theory
has traditionally focused on towers of number fields, the function field setting
has been studied extensively, and has important applications to the theory of `p`-adic modular forms.
This talk will introduce an exciting new kind of `p`-adic Iwasawa theory for towers of function fields over
finite fields of characteristic `p`, and discuss some applications to problems and open conjectures in the field.

### 20 May, 2022

Automorphic Forms, a Generalisation of Modular Forms, Part 1

Chenyan Wu (the University of Melbourne)

Notes taken by Chengjing Zhang

### 13 May, 2022

Some Applications of Modular Forms

Yiannis Fam (the University of Melbourne)

### 6 May, 2022

Geometric Aspects of Modular Forms

Alex Ghitza (the University of Melbourne)

### 27 Apr., 2022

Siegel Modular Forms

Chengjing Zhang (the University of Melbourne)

### 15 Apr. and 22 Apr., 2022

No talks due to Easter Holidays

### 8 Apr., 2022

Hecke Operators

Alex Ghitza (the University of Melbourne)

### 1 Apr., 2022

Modular Forms of Half Integral Weight

Alex Ghitza (the University of Melbourne)

### 25 Mar., 2022

A Recap on Modular Forms

Chenyan Wu (the University of Melbourne)

### 18 Mar., 2022

Algebraic Groups, Mostly Classical Groups

Chenyan Wu (the University of Melbourne)