# Number Theory Seminar @ University of Melbourne

## 11:00-12:00 Thu.

Alex Ghitza and I are running a working seminar on things number-theoretic. It runs at 11:00-12:00 Thu. (Melbourne time) this semester via Zoom. 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.

The current focus is modular forms and Siegel modular forms. We plan to follow largely the book, The 1-2-3 of Modular Forms by
Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder and Don Zagier. We aim to introduce students to modular forms and their generalisations. This should offer a nice pathway into arithmetic geometry and help us understand research topics such as congruences between automorphic forms on different groups.

The focus of 2021S1 is the global theory of Jacquet-Langlands Theory. We will make use of the local theory that we discussed last semester. We will recall the results when needed, so it is OK if you missed the first season (link containing seminar notes). We will continue to follow Notes on Jacquet-Langlands' Theory by Godement.

### 21 Oct., 2021

Singular Forms, Theta Series, and Fourier-Jacobi Expansions

Chenyan Wu (the University of Melbourne)

### 14 Oct., 2021

Eisenstein series in the Siegel setting

Alex Ghitza (the University of Melbourne)

### 7 Oct., 2021

Introduction to Siegel Modular Forms

Chenyan Wu (the University of Melbourne)

### 30 Sept., 2021

L-series (continued)

Alex Ghitza (the University of Melbourne)

### 16 Sept., 2021

Theta series

Simon Thomas (the University of Melbourne)

### 9 Sept., 2021

Hecke operators, Hecke eigenfunctions, and related L-series

Alex Ghitza (the University of Melbourne)

### 2 Sept., 2021

Examples of Modular Forms: Eisenstein Series and the Discriminant Function (Part 2)

Chenyan Wu (the University of Melbourne)

### 26 Aug., 2021

Modular Groups, Modular Functions, and Modular Forms (part 2)

Alex Ghitza (the University of Melbourne)

Examples of Modular Forms: Eisenstein Series and the Discriminant Function

Chenyan Wu (the University of Melbourne)

### 19 Aug., 2021

Modular Groups, Modular Functions, and Modular Forms

Alex Ghitza (the University of Melbourne)

### 12 Aug., 2021

Introduction to Modular Forms

Alex Ghitza (the University of Melbourne)

### 15 July, 2021

Eisenstein congruences for GL(2)

Alex Ghitza (the University of Melbourne)

It is possible for an Eisenstein series and a cusp form of the same weight to be congruent modulo a prime number. This phenomenon was first noticed by Ramanujan and its natural generalisations were subsequently explored by several authors. I will discuss some of these generalisations, following mostly the 2014 paper "Ramanujan-style congruences of local origin" by Dummigan and Fretwell.

### 8 July, 2021

Report on Lifting Puzzles and Congruences

Chenyan Wu (the University of Melbourne)

### 17 June, 2021

Archimedean Theory II

Last talk of the semester (2021S1)

Lance Gurney (the University of Melbourne)

### 10 June, 2021

Archimedean Theory I

Lance Gurney (the University of Melbourne)

### 3 June, 2021

The Converse Theorem

Chenyan Wu (the University of Melbourne)

### 20 May, 2021

Global L-functions II

Alex Ghitza (the University of Melbourne)

### 6 May, 2021

Global L-functions I

Alex Ghitza (the University of Melbourne)

### 29 Apr., 2021

Uniqueness of global Whittaker model and Multipicity One theorem for GL(2)

Chenyan Wu (the University of Melbourne)

### 22 Apr., 2021

Review of Admissible Representations and Whittaker Models

Chenyan Wu (the University of Melbourne)

### 15 Apr., 2021

Global Hecke Algebra

Simon Thomas (the University of Melbourne)

### 8 Apr., 2021

Easter Break. No talk.

### 1 Apr., 2021

Decompostion of Space of Cuspidal Functions; Tensor Product Decomposition

Alex Ghitza (the University of Melbourne)

### 25 Mar., 2021

Decomposition of Space of Cuspidal Functions: Compact Operators

Chenyan Wu (the University of Melbourne)

### 18 Mar., 2021

Adelic Groups and Cuspidal Functions for GL(2)

Chenyan Wu (the University of Melbourne)