# Number Theory Seminar @ University of Melbourne

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

Time: 12:00 - 13:00 Wed. (Melbourne local time) for 2023S2.

Venue: Peter Hall 162 (The 🌟NEW🌟 seminar room)

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.

Time: 12:00 - 13:00 Thu. (Melbourne local time) for 2023S1.

Venue: Peter Hall 162 (The 🌟NEW🌟 seminar room)

Topic: We continue our discussion on the book by Getz and Hahn, An Introduction to Automorphic Representations with a view toward Trace Formulae, since last semester. We plan to cover mainly Chapters 7, 11, 12 and possibly other parts.

### 25 Oct. 2023

No talk. End of Semester 2. We plan to resume in 2024S1.

### 18 Oct. 2023

Irreducible admissible representations of GL_{n}(F), where F is a p-adic field, part 2

### 11 Oct. 2023

Irreducible admissible representations of GL_{n}(F), where F is a p-adic field, part 1

I will describe the Bernstein--Zelevinsky classification of irreducible admissible representations of GL_{n}(F).
Time permitting, I will also describe the Langlands classification.

### 4 Oct. 2023

Local theta correspondence

Chengjing Zhang (The University of Melbourne)

In this talk, we will start with the local theta correspondence between two-fold coverings of algebraic groups over local fields and then discuss when the theta lifts are non-zero. Although in general we cannot embed the algebraic groups in a reductive dual pair into their two-fold coverings, for some of them we can embed them into their extensions by the circle group. Using such embeddings, the local theta correspondence induces a correspondence between irreducible representations of the algebraic groups whose theta lifts are non-zero.

### 27 Sept. 2023

No talk. Semester Break.

### 20 Sept. 2023

Values of the Ramanujan tau-function

Vandita Patel (The University of Manchester)

The infamous Ramanujan tau-function is the starting point for many mysterious conjectures and difficult open problems within the realm of modular forms. In this talk, I will discuss some of our recent results pertaining to odd values of the Ramanujan tau-function. We use a combination of tools which include the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue--Mahler equations due to Bugeaud and Gyory, and the modular approach via Galois representations of Frey--Hellegouarch elliptic curves. This is joint work with Mike Bennett (UBC), Adela Gherga (Warwick) and Samir Siksek (Warwick).

### 13 Sept. 2023

Classification of reductive dual pairs

Chengjing Zhang (The University of Melbourne)

In the overview of the theta correspondence, we stated the Howe duality theorem as a bijection between certain representations of two-fold coverings of the two groups in a reductive dual pair. In a lot of cases, this bijection induces a bijection between certain representations of the two groups themselves. To describe this, we need to classify reductive dual pairs in symplectic groups.

In this talk, we will start from Hermitian forms over algebras with involution and then classify reductive dual pairs over a general field. After that, we will briefly discuss quaternion algebras so that we can describe a more specific classification over local fields.

### 6 Sept. 2023

No talk. We will be attending the NTDU conference.

### 23 Aug. 2023

The Weil representations 2

Chengjing Zhang (The University of Melbourne)

This week, we will conclude our discussion about the Weil representations. We will start from the construction of the local and the global Weil representations and then describe the smooth representations corresponding to the Schrödinger representations and the Weil representations.

### 16 Aug. 2023

The Weil representations

Chengjing Zhang (The University of Melbourne)

Last week, we saw that the Stone–von Neumann theorem implies the existence of the Weil representation, but it could not provide us enough information about it. This week, we will use the automorphism group of the Heisenberg group to construct the Weil representation and derive its formula. After that, we will discuss how to obtain the local and adelic Weil representations.

### 9 Aug. 2023

The Heisenberg group and the Weil representation

Chengjing Zhang (The University of Melbourne)

The Weil representation, also called the oscillator representation, originates from the mathematical formulation of quantum mechanics. The representation-theoretic description of one-dimensional quantum mechanical systems involves certain representations of the real Heisenberg group, the group of 3×3 upper triangular real matrices with all diagonal entries 1. These representations can be classified by a theorem called the Stone–von Neumann theorem, which can then be used to construct the Weil representation of Mp(2,ℝ). The theory of the real Heisenberg group leads to a similar theory of the Heisenberg group of a locally compact Hausdorff Abelian group. The application of this theory to vector spaces provides the foundation of the theta correspondence.

In this talk, we will start from the classical theory of the real Heisenberg group. After that, we will review some results about locally compact Hausdorff Abelian groups and then discuss the general theory of Heisenberg groups.

### 2 Aug. 2023

Congruences related to modular forms

Sadiah Zahoor

The objective of this talk is to introduce and to prove congruences related to modular forms. Our main motivation to study such congruences is the fact that these can be used to give evidence in support of well known conjectures in Number Theory like Birch Swinnerton-Dyer conjecture and the Tamagawa conjecture of Bloch and Kato. In order to do so, we use the theory of half-integer weight modular forms and the arithmetic significance of their Fourier coefficients as captured by Waldpurger’s theorem.

We will show how mod p congruences between modular forms in general transcend to similar mod p congruences between half-integer weight modular forms with slightly shifted Fourier coefficients. We try to use a recipe in our proof that can be easily generalised to Hilbert modular forms over totally real quadratic fields of narrow class number 1.

### 26 July, 2023

The Theta correspondence

Chengjing Zhang (The University of Melbourne)

This is the first of several talks about the theta correspondence. The theta correspondence is a technique for transferring representations or automorphic forms from one group to another. In this talk, I will give an overview of both the local and the global theta correspondence.

### 1 June, 2023

No talk. We plan to resume in 2023 Semester 2 (which starts on 24 July.)

### 25 May, 2023

Ghitza's "elementary" problem

At 107 Peter Hall Building.
Note special location.

I will talk about an "elementary" problem involving quintics with a single real root, arising in the search for certain examples of residually modular abelian surfaces.

### 18 May, 2023

Rankin--Selberg Integrals and Rankin--Selberg L-functions

### 11 May, 2023

Multiplicity One Theorem for GL_{n}

### 3 May, 2023 (Wed.)

10:00-11:00, at 107 Peter Hall Building.
Note special time, day and location.

de Rham crystals on the prismatic site

Let `K` be a `p`-adic local field. We classify de Rham crystals over the absolute prismatic site of 𝒪_{K} using certain log-connections and certain nearly de Rham Galois representations. We also compare their various cohomologies. This is joint work with Min and Wang.

### 27 Apr. 2023

Whittaker models

### 20 Apr. 2023

Satake isomorphism for reductive groups over nonarchimedean local fields III

Generic Representations

### 13 Apr. 2023

Semester break. No seminar.

### 6 Apr. 2023

Cosets and Hecke operators

The goal is to describe what the Hecke operators are and how
they arise from cosets and double cosets. The groups that play a role
are primarily GL_{2}(ℚ) and SL_{2}(ℤ).

### 30 Mar. 2023

Satake isomorphism for reductive groups over nonarchimedean local fields II

### 23 Mar. 2023

Satake isomorphism for reductive groups over nonarchimedean local fields I

### 9 Mar. 2023

Unramified representations

### 2 Mar. 2023

A recap on automorphic representations