# 2023-24

A list of members of the group and research intersts is available here

The Number Theory Seminar takes place on Mondays from 3 pm to 4 pm (or on Tuesday if Monday is a bank holiday).

Seminars are held in B3.02 of the Zeeman building (MS.03 if Tuesday) and live-streamed on Teams; sometimes the speaker is online but the talk will still be streamed in B3.02. Sometimes a different room is used, see the entry below.

### 2023-24 Term 1

Organisers: Simon Myerson, Ju-Feng Wu and Harry Schmidt

 2 Oct Riccardo Maffucci (Coventry) Limiting Theorems for Arithmetic Eigenfunctions This is joint work with A. Rivera. Several recent papers study the ensemble of Laplace Toral eigenfunctions, and their randomisation 'arithmetic waves,' introduced in 2007 by Oravecz-Rudnick-Wigman, and by now a 'classical' setting. These waves are related to the arithmetic of writing a number as the sum of $d$ squares, where $d$ is the dimension. One question is the nodal volume of the wave, in the high-energy limit. Rivera and I considered a wider class of 'arithmetic fields,' eigenfunctions of a certain operator. Instead of a sum of squares, we work with (homogeneous) forms in a certain degree and dimension. If the dimension is much larger than the degree, we have a precise asymptotic for the variance of nodal volume, in the high-energy limit. We also prove the limiting distribution to be Gaussian. To solve the problem, we found the precise order of magnitude for the 'correlations,' i.e., tuples of lattice points summing up to $0$. In person 9 Oct Postponed Maria-Romina Ivan (Cambridge) Monochromatic Sums and Products over the Rationals Hindman’s Theorem states that whenever the natural numbers are finitely coloured there exists an infinite sequence all of whose finite sums are the same colour. By considering just powers of 2, this immediately implies the corresponding result for products: whenever the naturals are finitely coloured there exists a sequence all of whose products are the same colour. But what happens if we ask for both the sums and the products to all have the same colour? It turns out that this is not true: it has been known since the 1970s that there is a finite colouring of the naturals for which no infinite sequence has the set of all of its sums and products monochromatic. In this talk, we will investigate what happens to this question if we move from the naturals to a larger space such as the dyadic rationals, the rationals, or even the reals. Joint work with Neil Hindman and Imre Leader. In person 16 Oct Xenia Dimitrakopoulou (Warwick) Anticyclotomic $p$-adic $L$-functions for families of $U_n \times U_{n+1}$ I will report on current work in progress on the construction of anticyclotomic $p$-adic $L$-functions for Rankin--Selberg products. I will explain how by $p$-adically interpolating the branching law for the spherical pair $\left(U_n, U_n \times U_{n+1}\right),$ we can construct a $p$-adic $L$-function attached to cohomological automorphic representations of $U_n \times U_{n+1}$, including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this $p$-adic $L$-function interpolates the square root of the central $L$-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation. In person 23 Oct David Lilienfeldt (Leiden) The Gross-Zagier formula for generalized Heegner cycles In the 1980s, Gross and Zagier famously proved a formula equating on the one hand the central value of the first derivative of the Rankin-Selberg convolution $L$-function of a weight $2$ eigenform with the theta series of a class group character of an imaginary quadratic field, and on the other hand the height of a Heegner point on the corresponding modular curve. This equality was a key ingredient in the proof of the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rationals in analytic rank $0$ and $1$. Two important generalizations present themselves: to allow eigenforms of higher weight, and to further allow Hecke characters of infinite order. The former generalization is due to Shou-Wu Zhang. The latter one is the subject of this talk and requires the calculation of the Beilinson-Bloch heights of generalized Heegner cycles. This is joint work with Ari Shnidman. In person 30 Oct Rachel Newton (King's College London) Distribution of genus numbers of abelian number fields Let $K$ be a number field and let $L/K$ be an abelian extension. The genus field of $L/K$ is the largest extension of $L$ which is unramified at all places of $L$ and abelian as an extension of $K$. The genus group is its Galois group over $L$, which is a quotient of the class group of $L$, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions $L/K$ with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran. In person 6 Nov Victor Souza (Cambridge) The number of monochromatic solutions to multiplicative equation Abstract: Given an r-colouring of the integer interval [2, N], what is the minimum number of monochromatic solutions of the equation xy = z? We answer this question, obtained a sharp answer for 2, 3 or 4 colours. We also explore related questions and conjectures.   Joint work with L. Aragão, J. Chapman and M. Ortega. In person 13 Nov Manuel Hauke (York) On the connections between metric Diophantine approximation and Birkhoff sums for irrational rotations In the theory of metric Diophantine approximation, the existence of unusually large partial quotients destroys the possibility of many classical limit laws (such as Strong law of large numbers, Central Limit Theorem, ...). I will describe how this phenomenon can be applied to obtain precise bounds for the typical and worst-case oscillation of Birkhoff sums $\sum_{n=1}^N f(n\alpha), N \to \infty$ for almost every $\alpha$ where $f$ is a $1$-periodic function with discontinuities or singularities. Further, I will present a Khintchine-type behaviour of $\sum_{n=1}^N f(n\alpha)$ and the (non-)existence of limit laws that again resembles the behaviour known from metric Diophantine approximation. This is joint work with L. Fruehwirth. In person 20 Nov Alisa Sedunova (LIMS) The multiplication table constant and sums of two squares Let $r_1(n)$ be the number of integers up to $x$ that can be written as the square of an integer plus the square of a prime. We will show that its mean is asymptotic to $(\pi/2) x \log x$ minus a secondary term of size $x/(\log x)^{1+d+o(1)}$, where d is the multiplication table constant. Detailed heuristics suggest very precise asymptotic for the secondary term as well. In particular, our proofs imply that the main contribution to the mean value of $r_1(n)$ comes from integers with “unusual” number of prime factors, i.e, those with $\omega(n) \sim 2 \log \log x$ (for which $r_1(n) \sim (\log x)^{\log 4-1}$), where $\omega(n)$ is the number of district prime factors of $n$. This is a joint work with Andrew Granville and Cihan Sabuncu. In person 27 Nov Maria-Romina Ivan (Cambridge) Monochromatic Sums and Products over the Rationals Hindman’s Theorem states that whenever the natural numbers are finitely coloured there exists an infinite sequence all of whose finite sums are the same colour. By considering just powers of 2, this immediately implies the corresponding result for products: whenever the naturals are finitely coloured there exists a sequence all of whose products are the same colour. But what happens if we ask for both the sums and the products to all have the same colour? It turns out that this is not true: it has been known since the 1970s that there is a finite colouring of the naturals for which no infinite sequence has the set of all of its sums and products monochromatic. In this talk, we will investigate what happens to this question if we move from the naturals to a larger space such as the dyadic rationals, the rationals, or even the reals. Joint work with Neil Hindman and Imre Leader. In person 4 Dec Thomas Karam (Oxford) Title TBC In person Postponed Alexander Molyakov (ENS Paris) Title TBC In person