Number Theory

Monday 23rd June

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

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

Wednesday 25th June

Contributed talks (further details tbc)

Titles and Abstracts

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 $h$ .

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 $\mathrm{GL}(3)$ .