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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.

2022-23 Term 3

Organisers: Simon Myerson, Ju-Feng Wu and Han Yu

24th April

Oren Becker (Cambridge)

Character varieties of random groups

The space Hom(\Gamma,G) of homomorphisms from a finitely generated group \Gamma to a complex semisimple algebraic group G is known as the G-representation variety of \Gamma. We study this space when G is fixed and \Gamma is a random group in the few-relators model. That is, \Gamma is generated by k elements subject to r random relations of length L, where k and r are fixed and L tends to infinity.

More precisely, we study the subvariety Z of Hom(\Gamma,G), consisting of all homomorphisms whose images are Zariski dense in G. We give an explicit formula for the dimension of Z, valid with probability tending to 1, and study the Galois action on its geometric components. In particular, we show that in the case of deficiency 1 (i.e., k-r=1), the Zariski-dense G-representations of a typical \Gamma enjoy Galois rigidity.

Our methods assume the Generalized Riemann Hypothesis and exploit mixing of random walks and spectral gap estimates on finite groups.

Based on joint work with E. Breuillard and P. Varju.

In person





Rustam Steingart (Heidelberg)

Analytic Cohomology of Lubin–Tate $(\varphi,\Gamma)$-Modules

If $L$ is a non-trivial finite extension of $\mathbb{Q}_p$ there exist $L$-linear representations of the absolute Galois group $G_L$ which are not overconvergent. A sufficient condition to ensure overconvergence is $L$-analyticity. This makes it interesting to study analytic extensions of such modules or, more generally, analytic cohomology. Using $p$-adic Fourier theory we can, after passing to a large field extension of $L$, describe these cohomology groups in terms of an explicit "Herr-complex" which allows us to deduce finiteness and base change properties analogous to the results of Kedlaya–Pottharst–Xiao on continuous cohomology. We also obtain a variant of Shapiro's Lemma for Iwasawa-cohomology in certain cases. The above results form the technical foundations for studying an "analytic" variant of the local epsilon-isomorphism conjecture.

In person, MS.03

15th May

Caleb Springer (UCL)

Abelian varieties over finite fields and their groups of rational points

Abelian varieties are a generalization of elliptic curves, in the sense that they are smooth projective varieties whose rational points form an abelian group. In this talk, we will introduce and deploy algebraic machinery to describe phenomena for the groups of rational points of abelian varieties over finite fields. This machinery can be used to answer many questions. For example, building on work of Howe and Kedlaya, we show that every finite abelian group arises as the group of rational points of some (ordinary) abelian variety defined over the finite field with 2 elements. This is joint work with Stefano Marseglia.

In person, B3.03
22nd May

Ian Petrow (UCL)

A non-archimedean Petersson/Kuznetsov formula, the spectral large sieve, and subconvexity

The Petersson/Kuznetsov formula is a spectral summation device for classical automorphic forms that allows one to select forms according to the irreducible admissible unitary representation of $\mathrm{GL}_2(\mathbb{R})$ that they generate. We present a generalized Petersson/Kuznetsov formula where one may select automorphic forms according to the irreducible admissible unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ that they generate instead. In the case where one selects a single supercuspidal representation of $\mathrm{PGL}_2(\mathbb{Q}_p)$ with even conductor exponent (or, a pair of supercuspidal representations with odd conductor exponent) we present an elegant expression for corresponding Kloosterman sum. As applications, we present a spectral large sieve inequality for these families of automorphic forms, and also applications to cubic moments and subconvexity. Everything in this talk is joint work in progress with M.P. Young and Y. Hu.

In person
TUESDAY 30th May

Marta Benozzo (LSGNT)

On the canonical bundle formula in positive characteristic

An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. The first instance of such a formula is Kodaira’s canonical bundle formula for surfaces which admit a fibration with elliptic fibres. It describes the relation between the canonical bundles in terms of the singularities of the fibres and their j-invariants. For higher dimensional varieties over fields of characteristic 0, such formula has been generalised using variations of Hodge structures. Recently, the problem has been approached with techniques from the minimal model program. These methods can be used to prove a canonical bundle formula result in positive characteristic.

In person, MS.03

5th June

Chris Williams (Nottingham)

Classical symplectic families of eigensystems for GL(N)
Let p be a prime and N an integer prime to p. Let f be an eigenform of level $\Gamma_0(Np)$. For a given integer m, does there exist another eigenform g of level such that $f \equiv g \mod p^m$? For classical modular forms, which are automorphic forms for GL(2), the answer is yes. Even better, every such eigenform f can be deformed in a 1-dimensional p-adic family of eigenforms as we p-adically deform the weight (captured by the 'eigencurve'). This object studies the p-adic variation of systems of Hecke eigenvalues, rather than eigenforms, and has had profound consequences in Iwasawa theory and the Langlands program.
It is natural to ask if this holds more generally. I will describe recent joint work with Daniel Barrera and Andy Graham, where we consider the setting of symplectic automorphic forms on GL(N). In this case, it turns out the question is more subtle. For example, if $\pi$ is an automorphic representation of GL(4) of level N, there are 24 attached eigensystems at level Np. We conjecture that 8 of them deform in 2-dimensional families, 8 of them in 1-dimensional families, and 8 of them in no family at all. If time permits, I'll describe an application: we conjecture a p-refined analogue of a result of Friedberg—Jacquet, on the non-vanishing of global period integrals for GL(2n) over GL(n)xGL(n).
In person

12th June

Ila Varma (Toronto)

Malle's Conjecture for octic $D_4$-fields.
We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.

19th June

Chi-Yun Hsu (Lille)

On partially classical Hilbert modular forms

Let $F$ be a totally real field. A Hilbert modular form is a section of a modular sheaf, defined over the whole Hilbert modular variety associated to $F$, while a $p$-adic overconvergent form is defined only over a strict neighborhood of the ordinary locus. For each subset $I$ of the primes of $F$ above $p$, one has the intermediate notion of $I$-classical Hilbert modular forms by replacing ordinary by $I$-ordinary. Given an overconvergent Hecke eigenform $f$, we have the associated Galois representation $\rho$, which is well-known to be de Rham at $p$ when $f$ is classical. We expect that $\rho$ is $I$-de Rham when $f$ is $I$-classical. The idea is to $p$-adically deform $f$ in the weight direction of the complement of $I$, show that classical points are dense, and using the fact that the $I$-de Rham locus is closed when the $I$-Hodge Tate weights are fixed.

In person

26th June

Shreyasi Datta (Michigan)

Quantitative Simultaneous Approximations

In a recent ground-breaking work (arXiv:2105.13872), Beresnevich and Yang proved Khintchine's theorem in the set-up of simultaneous Diophantine approximation for nondegenerate manifolds, resolving a long-standing problem. In this talk, we will explain an effective version of their result.

This is based on the work


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

Nikoleta Kalaydzhieva (UCL)

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:

  1. 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);
  2. 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.


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