# Number Theory Seminar

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

In 2022-23, the seminar series has returned to in-person-only mode by default.

In-person seminars are held in B3.02 of the Zeeman building (MS.03 if Tuesday) and no longer live-streamed. There are occasional remote talks, which will be held on Microsoft Teams and broadcast in B3.02.

### 2022-23 Term 2

Organisers: Pak-Hin Lee, Simon Myerson and Han Yu

 9th January Wissam Ghantous (Oxford) A symmetric triple product p-adic L-function In 2014, Darmon and Rotger define the Garrett–Rankin triple product p-adic L-function and relate it to the image of certain diagonal cycles under the p-adic Abel–Jacobi map. We introduce a new variant of this p-adic L-function and show that it satisfies symmetry relations, when permuting the three families of modular forms. We also provide computational evidence confirming that it is indeed cyclic when the families of modular forms are evaluated at even weights, and provide counter-examples in the case of odd weights. To do so, we extend Lauder's algorithm (for computing ordinary projections of nearly overconvergent modular forms) to work with nearly overconvergent modular forms and compute projections over spaces of non-zero slope. In person 16th January Julia Stadlmann (Oxford) The mean square gap between primes Conditional on the Riemann hypothesis, Selberg showed in 1943 that the average size of the squares of differences between consecutive primes less than x is O(log(x)^4). Unconditional results still fall far short of this conjectured bound: Peck gave a bound of O(x^{0.25+epsilon}) in 1996 and to date this is the best known bound obtained using only methods from classical analytic number theory. In this talk we discuss how sieve theory (in the form of Harman's sieve) can be combined with classical methods to improve bounds on the number of short intervals which contain no primes, thus improving the unconditional bound on the mean square gap between primes to O(x^{0.23+epsilon}). In person 23rd January Elisa Lorenzo García (Neuchâtel) Lower bound on the maximal number of rational points on curves over finite fields For a long time people have been interested in finding and constructing curves with many points. For genus 1 and genus 2 curves, we know how to construct curves over any finite field of defect less than 1 or 3 (respectively), i.e. with a number of points at distance at most 1 or 3 to the upper bound given by the Hasse–Weil–Serre bound. The case of genus 3 is still open after more than 40 years of research. In this talk I will take a different approach based on the random matrix theory of Katz–Sarnak to prove the existence, for all \epsilon > 0, of curves of genus g over Fq with more than 1 + q + (2g −\epsilon )\sqrt{q} points for q big enough. I will also discuss some explicit constructions as well as some consequences to the Serre obstruction problem (an asymmetric behaviour of the distribution of the trace of the Frobenius for curves of genus 3). This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler. Teams, Oculus OC1.08 30th January Alex Walker (UCL) Sums of Hecke Eigenvalues in a Quadratic Sequence Many arithmetic functions which are well-understood on average over sets of positive density remain mysterious when considered over sparser sets. For example, it is not known if there are infinitely many primes of the form n^2 + 1. The behavior of the divisor function on quadratic sequences was first studied by Hooley and refined by Bykovskii. More recently, Blomer has asked a similar question for the Hecke eigenvalues of a holomorphic cusp form. In this talk, we show how to strengthen Blomer's error estimate through the use of shifted convolution sums and the spectral theory of (half-integral weight) automorphic forms. In person 6th February Amitay Kamber (Cambridge) Optimal Lifting for SL_n(Z) Let $q$ be a natural number. The strong approximation theorem for $SL_n(\mathbb{Z})$ says that the modulo $q$ map $SL_n(\mathbb{Z}) \to SL_n(\mathbb{Z}/q\mathbb{Z})$ is onto. This leads to the following research problem: Given a parameter $T$, look at the (finite) set of matrices $B_T := \{ A\in SL_n(Z) : ||A|| \le T \}$, where $||.||$ is some matrix norm. We are interested in understanding the image of $B_T$ in $SL_n(\mathbb{Z}/q\mathbb{Z})$, for $T$ a function of $q$. Such studies were initiated (in a more general context) by Duke–Rudnick–Sarnak, and further developed by many others, notably Gorodnik–Nevo. We will focus on the problem of covering $SL_n(\mathbb{Z}/q\mathbb{Z})$ with the image of $B_T$, and explain the connection of the problem to the Generalized Ramanujan Conjecture in automorphic forms. Based on a joint work with Subhajit Jana. In person 13th February Properties of the multiple solutions to the polynomial Pell equation In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell’s equation for $D$. It is well-known that, once an integer solution to Pell’s equation exists, we can use it to generate all other solutions $(u_n,v_n)$ ($n\in\mathbb{Z}$). Our object of interest is the polynomial version of Pell’s equation, where the integers are replaced by polynomials with complex coefficients. We investigate the factors of $v_n(t)$. In particular, we show that over the complex polynomials, there are only finitely many values of n for which $v_n(t)$ has a repeated root. Restricting our analysis to $\mathbb{Q}[t]$, we give an upper bound on the number of “new” factors of $v_n(t)$ of degree at most $N$. Furthermore, we show that all “new” linear rational factors of $v_n(t)$ can be found when $n \leq 3$, and all “new” quadratic rational factors when $n \leq 6$. In person 20th February Sacha Mangerel (Durham) An explicit construction of multiplicative functions with small correlations A conjecture of Chowla, analogising the Hardy-Littlewood prime $k$-tuples conjecture, predicts that the autocorrelations of $\lambda$ (the completely multiplicative function taking the value -1 at all primes) tend to 0 on average, e.g., $\frac{1}{x}\sum_{n \leq x} \lambda(n+1)\cdots \lambda(n+k) \rightarrow 0 \text{ as } x \rightarrow \infty$. This conjecture, along with its generalisation to other bounded non-pretentious'' multiplicative functions due to Elliott, remain wide open for $k \geq 2$. In this talk I will present an explicit construction of a non-pretentious multiplicative function $f: \mathbb{N} \rightarrow \{-1,1\}$ all of whose auto-correlations tend to 0 on average, answering a question of Lemanczyk. I will further discuss the following applications of this construction: a proof that Chowla's conjecture does not imply the Riemann Hypothesis, i.e., there are functions $f$ all of whose autocorrelations tend to 0, but that do not exhibit square-root cancellation on average (the object of some recent speculation); there are multiplicative subsemigroups of $\mathbb{N}$ with Poissonian gap statistics, thus giving an unconditional multiplicative analogue of a classical result of Gallagher about primes in short intervals. (Joint work with Oleksiy Klurman, Pär Kurlberg and Joni Teräväinen) In person 27th February Efthymios Sofos (Glasgow) Schinzel's Hypothesis on average and the Hasse principle Schinzel's Hypothesis regards prime values taken by a polynomial with integer coefficients. I will talk about work with Skorobogatov where we established that the Hypothesis holds for 100% of cases in a probabilistic sense. I will also talk about joint work with Browning and Teräväinen where we extend the previous result in various directions. In person 6th March Peiyi Cui (East Anglia) Decompositions of the category of l-modular representations of SL_n(F) Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with respect to several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A. In person 13th March Cécile Dartyge (Lorraine) On the largest prime factor of quartic polynomial values : the cyclic and dihedral cases. Let $P(X)$ be an irreducible, monic, quartic polynomial with integral coefficients and with cyclic or dihedral Galois group. There exists $c_P >0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor bigger than $n^{1+c_p}$. This is a joint work with James Maynard. In person

### 2022-23 Term 1

Organisers: Pak-Hin Lee, Oscar Rivero Salgado and Han Yu

 3rd October Matteo Tamiozzo (Warwick) Perfectoid quaternionic Shimura varieties and the Jacquet–Langlands correspondence The Hodge–Tate period map can be thought of as a p-adic analogue of the Borel embedding. However, unlike its complex counterpart, it is not injective, and the pushforward of the constant sheaf via the Hodge–Tate period map encodes interesting arithmetic information. In the setting of quaternionic Shimura varieties, I will explain the relation between the structure of this complex of sheaves and level raising and the Jacquet–Langlands correspondence. I will then discuss applications to the study of the cohomology of quaternionic Shimura varieties. I will illustrate most of the arguments in the simplest setting of modular and Shimura curves. This is joint work with Ana Caraiani. In person 10th October Lambert A'Campo (Oxford) Galois representations and cohomology of congruence subgroups In this talk I will explain what it means to attach Galois representations to the cohomology of arithmetic locally symmetric spaces arising from congruence subgroups. In the case of GL(2) over imaginary CM fields (the method also works for GL(n)) I will explain how to prove, under certain conditions, that the Galois representations constructed by Harris–Lan–Taylor–Thorne and Scholze have good p-adic Hodge theoretic properties. In person 17th October Maria Rosaria Pati (Padova) L-invariants for cohomological representations of PGL(2) over an arbitrary number field In this talk I will construct the automorphic L-invariant attached to a cuspidal representation $\pi$ of PGL(2) over an arbitrary number field F, and a prime $\mathfrak{p}$ of F such that the local component $\pi_\mathfrak{p}$ is the Steinberg representation and $\pi$ is non-critical at $\mathfrak{p}$. I will show that, if F is totally real then the automorphic L-invariant attached to $\pi$ and $\mathfrak{p}$ agrees with the derivatives of the $U_\mathfrak{p}$-eigenvalue of the p-adic family passing through $\pi$. From this I will deduce the equality between the automorphic L-invariant and the Fontaine-Mazur L-invariant of the associated Galois representation. This is a joint work with Lennart Gehrmann. In person 24th October Aleksander Horawa (Oxford) Motivic action on coherent cohomology of Hilbert modular varieties A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties. In person, Science Concourse B2.04/05 31st October Yoav Gath (Cambridge) Lattice point statistics for Cygan–Koranyi balls Euclidean lattice point counting problems, the classical example of which is the Gauss circle problem, are an important topic in classical analysis and have been the driving force behind much of the developments in the area of analytic number theory in the 20th century. In this talk, I will introduce the lattice point counting problem for (2q+1)-dimensional Cygan–Koranyi balls, namely, the problem of establishing error estimates for the number of integer lattice points lying inside Heisenberg dilates of the unit ball with respect to the Cygan–Koranyi norm. I will explain how this problem arises naturally in the context of the Heisenberg groups, and how it relates to the Euclidean case (and in particular to the Gauss circle problem). I will survey some of the major results obtained to date for this lattice point counting problem, and in particular, results related to the fluctuating nature of the error term. In person TUESDAY 8th November George Boxer (Imperial) Higher Hida theory for Siegel modular varieties The goal of higher Hida theory is to study the ordinary part of coherent cohomology of Shimura varieties integrally. We introduce a higher coherent cohomological analog of Hida's space of ordinary p-adic modular forms, which is defined as the ordinary part of the coherent cohomology with "partial compact support" of the ordinary Igusa variety. Then we give an analog of Hida's classicality theorem in this setting. This is joint work with Vincent Pilloni. In person, MS.03 14th November Rosa Winter (KCL) Density of rational points on del Pezzo surfaces of degree 1 Let $X$ be an algebraic variety over an infinite field $k$. In arithmetic geometry we are interested in the set $X(k)$ of $k$-rational points on $X$. For example, is $X(k)$ empty or not? And if it is not empty, is $X(k)$ dense in $X$ with respect to the Zariski topology? Del Pezzo surfaces are surfaces classified by their degree $d$, which is an integer between 1 and 9 (for $d\geq3$, these are the smooth surfaces of degree $d$ in $\mathbb{P}^d$). For del Pezzo surfaces of degree at least 2 over a field $k$, we know that the set of $k$-rational points is Zariski dense provided that the surface has one $k$-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field $k$, even though we know that they always contain at least one $k$-rational point, we do not know if the set of $k$-rational points is Zariski dense in general. I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of $k$-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where $k$ is finitely generated over $\mathbb{Q}$. In person 21st November Rachel Greenfeld (IAS) Aperiodicity of translational tilings Translational tiling is a covering of a space using translated copies of some building blocks, called the "tiles", without any positive measure overlaps. What are the possible ways that a space can be tiled? A well known conjecture in this area is the periodic tiling conjecture, which asserts that any tile of Euclidean space admits a periodic tiling. In a joint work with Terence Tao, we construct a counterexample to this conjecture. In the talk, I will survey the study of the periodicity of tilings and discuss our recent progress. Teams 28th November Alexandre Maksoud (Paderborn) The arithmetic of the adjoint of a weight 1 modular form A conjecture of Darmon, Lauder and Rotger expresses p-adic iterated integrals attached to a pair of weight 1 modular forms (f,g) in terms of p-adic logarithms of certain units attached to f and g. This talk reports a work in progress in which we explain, in the case where f=g, how to interpret this conjecture as a variant of the Gross-Stark conjecture for the adjoint of f. This requires studying the specializations of the congruence module attached to a Hida deformation of f. In person 5th December Istvan Kolossvary (St Andrews) Distance between natural numbers based on their prime signature One can define different metrics between natural numbers based on their unique prime signature. Fixing such a metric, we are interested in the asymptotic growth rate of the arithmetic function $L(N)$ which tabulates the cumulative sum of distances between consecutive natural numbers up to $N$. In particular, choosing the maximum norm, we will show that the limit of $L(N)/N$ exists and is equal to the expected value of a certain random variable. We also demonstrate that prime gaps exhibit a richer structure than on the traditional number line and pose a number of problems. Joint work with Istvan B. Kolossvary. In person