Number Theory

Monday 23rd June, Newman Red (WSNT1)

15:10 – 15:50 Andrew Pearce-Crump (Bristol)
15:50 – 16:30 Jesse Pajwani (Bath)
16:30 – 17:10 Chris Williams (Nottingham)

Tuesday 24th June, Newman Red (WSNT2)

15:40 – 16:20 Celine Maistret (Bristol)
16:20 – 17:00 Natalie Evans (KCL)
17:00 – 17:40 Robert Kurinczuk (Sheffield)

Wednesday 25th June, Newman Red (WSNT3)

15:40 – 16:00 Thomas Karam (Oxford)
16:05 – 16:25 Madhuparna Das (Exeter)
16:30 – 16:50 Zain Kapadia (Queen Mary)
16:55 – 17:15 David Kurniadi Angdinata (UCL)
17:20 – 17:40 Harvey Gray and Hollie Davis (Exeter)

Titles and Abstracts

David Kurniadi Angdinata – Elliptic divisibility sequences in Lean
Elliptic divisibility sequences are integer sequences satisfying a certain cyclic recurrence relation, which are crucial in the algebraic theory of division polynomials used in elliptic curve cryptography algorithms. In this talk, I will explain their construction and our subsequent formalisation in the Lean theorem prover, including how we inadvertently discovered a massive gap in the literature. This is joint work with Junyan Xu.

Madhuparna Das – Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions
In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$ , where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values, which has recently been generalized by Bridges et al. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs.

Natalie Evans – Correlations of almost primes
The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\leq X$ such that $p+h$ is prime, where $h$ is any non-zero even integer. While this conjecture remains wide open, Matomäki, Radziwiłł and Tao proved that it holds on average over $h$ , improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \leq X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$ .

Harvey Gray and Hollie Davis – Employment of AI on pure mathematics education
The artificial intelligence sector has been proven on numerous occasions to be a viable way of aiding students’ educational experience by serving as a digital personal tutor. Our presentation investigates to what effect ChatGPT can have on pure mathematics education. This presentation investigates ChatGPT’s fluency with LEAN. LEAN is an interactive theorem prover that enables mathematical proof to be written and interpreted on a computer. LEAN has been credited to be a breakthrough in the future of pure mathematics education and is gradually being enrolled into the curriculum of mathematics programs at universities. In the initial stages of primary research, we challenge ChatGPT with different pure mathematics problems in the realm of propositional logic, a topic that beginning mathematics undergraduates are introduced to. We then investigate the LEAN solutions it provides and verify their validity. This leads to a firm understanding of ChatGPT’s fluency with LEAN, allowing us to understand the challenges that are associated with developing a CustomGPT that will help students write mathematical proof in LEAN. Rigorous development and testing is performed on the CustomGPT in the form of flowcharts and written instructions in order to assess its capability and its limitations. From this process, we conclude to what extent the CustomGPT can act as a digital personal tutor for the students’ education in pure mathematics.

Zain Kapadia – Some Uniserial Specht Modules
The Representation Theory of the Symmetric Groups is a classical and rich area of combinatorial representation theory. Key objects of study include Specht modules, the irreducible ordinary representations, which can be reduced modulo $p$ (for $p$ prime). In general, these are no longer irreducible and finding their decomposition numbers and submodule structures are key questions in the area. We give sufficient and necessary conditions for a Specht module in characteristic $2$ , labelled by a hook partition to be a direct sum of uniserial summands.

Thomas Karam – The amount and type of structure of a function in terms of its small images
It is well known that if $G$ is an abelian group and $A$ is a finite subset of $G$ which minimises the size of $A+A$ for a fixed size of $A$ , then $A$ must essentially be as close as possible to being a coset of some subgroup of $G$ : for instance, if $G$ is finite then the set $A$ must be a coset in $G$ (provided that $G$ has subgroups with size $\lvert A \rvert$ ), and if $G=\mathbb{Z}$ then the set $A$ must be an arithmetic progression. One may wish to describe an extension of this phenomenon to more arbitrary functions than addition on abelian groups. If $X$ is a set and $F: X \times X \rightarrow X$ is a function, then can the structure of $F$ (to the extent that there is some) be read from the structure of the subsets $A$ of $X$ minimising the size of the image $F(A,A)$ for a fixed size of A, and can the “amount of structure” of $F$ be read from how quickly that minimal size grows with the size of $A$ ? We will discuss the progress that has been made in this direction, starting with the results of Elekes-Ronyai, and what remains to be understood.

Robert Kurinczuk – Finiteness for Hecke algebras of p-adic groups
Let $G$ be a $p$ -adic group and $K$ a compact open subgroup of $G$ . Over forty years ago, Bernstein proved that the Hecke algebra $\mathbb{C}[K\backslash G/K]$ is a finitely generated module over its centre and that its centre is a finitely generated $\mathbb{C}$ -algebra. It has long been expected that analogues of Bernstein’s result should hold over other coefficient rings. In joint work with Dat, Helm, and Moss, we established such analogues of Bernstein’s finiteness results after inverting $p$ . The key ingredient which unlocked the problem was Fargues and Scholze’s geometrization of the local Langlands correspondence. We will explain some of the main ideas from this approach, and time permitting discuss some current work in progress on Local Langlands in Families.

Celine Maistret – Galois module structure of Tate-Shafarevich groups of elliptic curves
For an elliptic curve $E/ \mathbb{Q}$ , computing its Tate-Shafarevich group is a fundamental but hard problem. Assuming its finiteness, the Birch and Swinnerton-Dyer conjecture (BSD) provides a formula for its size in terms of the special value of the L-function of $E/\mathbb{Q}$ and some arithmetic invariants of the curve. If $K/\mathbb{Q}$ is a Galois extension, one can ask if a BSD-like formula can be used to link the Galois module structure of the Tate-Shafarevich group of $E/K$ to the special value of the twisted $L$ -function of $E/\mathbb{Q}$ by an Artin representation associated to $K/\mathbb{Q}$ . In this talk, I will present a joint work with Himanshu Shukla, where we give an explicit conjecture for such a link in the case of primitive Dirichlet characters and provide numerical verifications obtained via 11-descent procedures over $C_5$ -extensions.

Jesse Pajwani – Symmetric powers of trace forms
Fix $k$ to be a field of characteristic not $2$ and let $L/k$ be a finite separable field extension. The Galois trace map ${\rm Tr}\colon L \rightarrow k$ gives rise to a quadratic form ${\rm Tr}_L$ on $L$ , viewed as a $k$ vector space, given by sending $x$ to ${\rm Tr}(x^2)$ . This construction extends to finite étale algebras over $k$ , so it is a natural question to ask whether this is compatible with operations on finite étale algebras over $k$ . In particular, if $A$ is a finite étale algebra, we can form the symmetric product $A^{(n)}$ , which is also a finite étale algebra. In this talk, we investigate the notion of a “symmetric power of quadratic forms” and show that ${\rm Tr}_{A^{(n)}}$ is given by a strange symmetric power of ${\rm Tr}_A$ , using results by Garibaldi-Merkurjev-Serre. If there is time at the end, we may discuss a higher dimensional analogue of this question, which concerns an arithmetic enhancement of the Euler characteristic of a variety. This is joint work with Ambrus Pál.

Andrew Pearce-Crump – Number Theory versus Random Matrix Theory: the joint moments story
It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height $T$ , is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$ . This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

Chris Williams – The exceptional zero conjecture for $\mathbf{GL}(3)$
The BSD conjecture predicts that a rational elliptic curve $E$ has infinitely many points if and only if its $L$ -function vanishes at $s=1$ . Iwasawa theory is concerned with $p$ -adic analogues of this phenomena. If $E$ is $p$ -ordinary, there is, for example, a $p$ -adic analytic analogue $L_p(E,s)$ of the $L$ -function; and if $E$ has good reduction at $p$ , then $L_p(E,1)=0$ if and only if $L(E,1)=0$ . However if $E$ has split multiplicative reduction at $p$ , then $L_p(E,1)=0$ for trivial reasons, regardless of $L(E,1)$ ; it has an ‘exceptional zero’. Mazur-Tate-Teitelbaum’s ‘exceptional zero conjecture’ (proved by Greenberg–Stevens in ’93) states that in this case the first derivative $L_p'(E,1)=0$ if and only $L(E,1)=0$ . They used this to formulate a precise $p$ -adic BSD conjecture. In this talk I will discuss exceptional zero phenomena, and describe joint work in progress with Daniel Barrera and Andy Graham proving the exceptional zero conjecture for $\mathrm{GL}(3)$ .