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Number Theory Seminar

Unless otherwise specified, the seminars are held on Mondays at 15:00 in Room B3.03 – Mathematics Institute.

Regular seminars have been suspended in light of the ongoing COVID-19 pandemic. We will be running seminars on Zoom during this time.

See [ this link ] for guidance/suggestions for using Zoom.

If you'd like to attend a seminar but don't have the meeting link, please email one of the organisers in advance.

2019-20 Term 3

Organisers: Sam Chow, Chris Lazda and Chris Williams

30th March

David Loeffler (Warwick)

On the Bloch-Kato conjecture for Siegel modular forms

The Bloch--Kato conjecture is a very broad conjecture relating the special values of L-functions to the arithmetic of Galois representations, generalising many familiar conjectures such as the Birch--Swinnerton-Dyer conjecture for elliptic curves. Recently, Sarah Zerbes and I succeeded in proving this conjeture for L-functions arising from Siegel modular forms fo Sp(4,Z), in analytic rank 0 (i.e. when the L-value doesn't vanish). I will explain what the conjecture says in this particular case, and try to give some flavour of the ideas involved in the proof.

20th April

Pol van Hoften (King's College London)

Title: Mod p points on Shimura varieties of parahoric level

Abstract: The conjecture of Langlands-Rapoport gives a conjectural description of the mod p points of Shimura varieties, with applications towards computing the (semi-simple) zeta function of these Shimura varieties. The conjecture was proven by Kisin for abelian type Shimura varieties at primes of (hyperspecial) good reduction, after having constructed smooth integral models. For primes of (parahoric) bad reduction, Kisin and Pappas have constructed 'good' integral models and the conjecture naturally generalises to this setting. In this talk we will discuss work in progress towards the conjecture for these integral models, under some hypotheses, building on earlier work of Zhou. I will try to make the talk fairly accessible, by extensively discussing the case of the modular curve first.

27th April

Arthur-Cesar le Bras (Institut Galilée, Univ. Paris XIII)

Prismatic Dieudonné theory

I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.

4th May

Pak-Hin Lee (Warwick)

A p-adic L-function for non-critical adjoint L-values

Let f be a classical eigenform, and K be an imaginary quadratic field with associated quadratic character \alpha. By works of Hida and Tilouine--Urban, the value L(1, ad(f) \otimes \alpha), which is non-critical in the sense of Deligne, measures congruences between f and (non-base-change) Bianchi modular forms over K. In this talk, we will outline the construction of an analytic p-adic L-function interpolating these special values as f varies in a Hida family. Our approach is based on Greenberg--Stevens' idea of \Lambda-adic modular symbols, which considers cohomology with values in a space of p-adic measures.

11th May

Rachel Newton (Reading)

Arithmetic of rational points and zero-cycles on Kummer varieties

In 1970, Manin observed that the Brauer group Br(X) of a variety X over a number field K can obstruct the Hasse principle on X. In other words, the lack of a K-point on X despite the existence of points over every completion of K is sometimes explained by non-trivial elements in Br(X). This so-called Brauer-Manin obstruction may not always suffice to explain the failure of the Hasse principle but it is known to be sufficient for some classes of varieties (e.g. torsors under connected algebraic groups) and conjectured to be sufficient for rationally connected varieties and K3 surfaces. A zero-cycle on X is a formal sum of closed points of X. A rational point of X over K is a zero-cycle of degree 1. It is interesting to study the zero-cycles of degree 1 on X, as a generalisation of the rational points. Yongqi Liang has shown that for rationally connected varieties, sufficiency of the Brauer-Manin obstruction to the Hasse principle for rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the Hasse principle for zero-cycles of degree 1 over K. In this talk, I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.

18th May

Shu Sasaki (Queen Mary)

Serre's conjecture about weight of mod p modular forms: old conjectures, not so old theorems and new conjectures

In 1987, J.-P. Serre made a set of conjectures about weights and levels of two-dimensional (modular) mod p Galois representations of the absolute Galois group of Q. This conjecture of Serre has been completely proved by C. Khare and J.-P. Wintenberger (2009) building on the work of many mathematicians, but it has also inspired a good deal of new mathematics.

One strand of research spurred on by the development is about generalising Serre's conjecture over to a (general) totally real number field. This was initiated by the work (2009) of K. Buzzard, F. Diamond and F. Jarvis, while focusing exclusively on regular weights of mod p Hilbert modular forms. In my joint work with F. Diamond, we have improved on the Buzzard-Diamond-Jarvis conjectures and formulated new conjectures about general weights of (geometric) mod p Hilbert modular forms (analogous to what B. Edixhoven did in 1992). I will explain what our conjectures say exactly, and demonstrate some evidence that we are on the right track. In support of our vision, I will also explain a comparatively new result (joint work with F. Diamond and P. Kassaei) about a Jacquet-Langlands relation between mod p geometric `Hilbert modular forms', which may well shed some light on the problem of formulating a putative mod p Langlands philosophy.

26th May (note date)

Steve Lester (Queen Mary)

Quantum variance for dihedral Maass forms

Motivated by connections with mathematical physics, a major topic within the analytic theory of automorphic forms is the distribution of their L^2-mass. One problem in this direction is to compute the quantum variance, which describes how far away the L^2-mass of a typical form is from being equidistributed. In this talk, I will describe some recent joint work with Bingrong Huang (Shandong), in which we compute the quantum variance over the family of dihedral Maass forms. In particular, the leading order constant in our formula for the quantum variance includes a geometric factor, which is consistent with a prediction from the physics literature, as well as a subtle arithmetic factor.

1st June

Minhyong Kim (Warwick)

Fragments of arithmetic topological field theories

There is an old analogy between knots and primes going back to the 1960s attributed to Mazur. I will outline some recent attempts to employ ideas from physics and geometry to connect it to the more 'usual' arithmetic geometry of Galois representations and their moduli.

8th June

Stephanie Chan (UCL)

Integral points on the congruent number curve

I will discuss some results on the upper bound of the number of integral points on the quadratic twists of the congruent number curve y^2=x^3-D^2 x, in relation to simultaneous Pell-like equations of a certain form. The proof involves a careful application of Roth's theorem similar to that in previous work of Alpoge.

15th June

Damaris Schindler (Goettingen)

On the distribution of Campana points on toric varieties
In this talk we discuss joint work with Marta Pieropan on the distribution of Campana points on toric varieties. We discuss how this problem leads us to studying a generalised version of the hyperbola method, which had first been developed by Blomer and Bruedern. We show how duality in linear programming is used to interpret the counting result in the context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.

22nd June

Daniel Barrera (Universidad de Santiago a Chile)

Families of automorphic p-adic L-functions and overconvergent cohomology  
Some important problems in arithmetic are conjecturally related to special values of the L-function attached to a certain automorphic representation. A fruitful procedure in this situation has been to p-adically deform the automorphic representation together with its L-values. In this talk I will explain a strategy to perform this p-adic deformation, it is based on the study of the overconvergent cohomology of arithmetic manifolds introduced by Ash-Stevens, Urban and Hansen, and a parahoric modification of it introduced in a joint work with C. Williams. Then, I will explain the situation in the case of GL2n where this strategy was carried out successfully in a joint work with M. Dimitrov and C. Williams.

2019-20 Term 2

Organisers: Sam Chow, Chris Lazda and Chris Williams

6th January

Mattia Sanna (Warwick)

A 3-adic Faltings-Serre method

Determining Galois representations, and proving isomorphisms between them, has proved crucial in modern number theory. One of the strongest tools we have for this is the Faltings-Serre-Livné method for two dimensional Galois representations that take values in a finite extension of \mathbb{Q}_2, and there has been extensive effort to convert these theoretical results into a deterministic and implementable algorithm. Ideally we would like to have an effective Faltings-Serre that works for a general n-dimensional Galois representation with values in a general local field. In this seminar I will discuss the main ideas and results that lead to an effective Faltings-Serre method for two dimensional Galois representations with values in \mathbb{Q}_3, and time permitting, its connection with the most recent result on modularity lifting due to Allen et al.(2019). The implementation is joint work with John Cremona.
13th January

Tom Oliver (Oxford)

Euler products, meromorphic continuation and twisting

We will discuss the interplay between the three themes of the title, giving applications to Artin and automorphic L-functions.

20th January

Christopher Frei (Manchester)

Abelian fields with prescribed norms

We discuss quantitative results concerning extensions K/k of a given number field k, with given abelian Galois group G, in which finitely many given elements of k can be realised simultaneously as norms from K. This is joint work with Dan Loughran and Rachel Newton, and with Rodolphe Richard.

27th January

Joni Teräväinen (Oxford)

Higher order uniformity of the Möbius function

In a recent work, Matomäki, Radziwill and Tao showed that the Möbius function is discorrelated with linear exponential phases on almost all short intervals. I will discuss joint work where we generalize this result to "higher order" phase functions, so as a special case the Möbius function is shown not to correlate with polynomial phases on almost all short intervals. As an application of this, we show that the Liouville sequence has superpolynomial subword complexity.
3rd February

Tobias Berger (Sheffield)

Irreducibility of limits of Galois representations

We prove (under certain assumptions) the irreducibility of the limit of a sequence of irreducible p-adic Galois representations which are residually reducible. This is recent joint work with Kris Klosin (CUNY) related to our work on the modularity of abelian surfaces which have a rational torsion point.

10th February

Federico Bambozzi (Oxford)

Analytic spaces over Banach rings

A well-known constrain of analytic geometry is that its foundations require to work with algebras of convergent power-series defined over quite specific Banach rings, like a valued fields for example. This is in stark contrast to algebraic geometry that works over any ring. Motivated by applications in arithmetic, in this talk I explain how the theory of Koszul complexes can be used to improve upon known results about the sheafyness of the structural sheaf of analytic functions on the adic spectrum of a Banach ring. This is a work in progress with Kobi Kremnizer.

17th February

Faustin Adiceam (Manchester)
Effective Equidistribution in Tori and Geometric Discrepancy
In connection with the Danzer Problem in Convex Geometry, we introduce the concept of strongly uniformly dispersed sequences in the torus. This is related to the concept of well-distributed sequences from discrepancy theory. We then show that there do exist sequences satisfying this property and sketch how this can be applied to construct so-called dense forests.

24th February

Ana Caraiani (Imperial)

Local-global compatibility in the crystalline case

Let F be a CM field. Scholze constructed Galois representations associated to classes in the cohomology of locally symmetric spaces for GL_n/F with p-torsion coefficients. These Galois representations are expected to satisfy local-global compatibility at primes above p. Even the precise formulation of this property is subtle in general, and uses Kisin’s potentially semistable deformation rings. However, this property is crucial for proving modularity lifting theorems. I will discuss joint work with J. Newton, where we establish local-global compatibility in the crystalline case under mild technical assumptions. This relies on a new idea of using P-ordinary parts, and improves on earlier results obtained in joint work with P. Allen, F. Calegari, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne in certain Fontaine-Laffaille cases.

2nd March

Vandita Patel (Manchester)

Shifted powers in Lucas-Lehmer sequences
The explicit determination of perfect powers in (shifted) non-degenerate, integer, binary linear recurrence sequences has only been achieved in a handful of cases. In this talk, we combine bounds for linear forms in logarithms with results from the modularity of elliptic curves defined over totally real fields to explicitly determine all shifted powers by two in the Fibonacci sequence.

9th March

Demi Allen (Bristol)

Dyadic approximation in the middle-third Cantor set
Motivated by a classical question due to Mahler, in 2007 Levesley, Salp, and Velani showed that the Hausdorff measure of the set of points in the middle-third Cantor set which can be approximated by triadic rationals (that is, rationals which have denominators which are powers of 3) at a given rate of approximation satisfies a zero-full dichotomy. More precisely, the Hausdorff measure of the set in question is either zero or full according to, respectively, the convergence or divergence of a certain sum which is dependent on the specified rate of approximation. Naturally, one might also wonder what can be said about dyadic approximation in the middle-third Cantor set. In this talk I will discuss a conjecture on this topic due to Velani, some progress towards this conjecture, and why dyadic approximation is harder than triadic approximation in the middle-third Cantor set. This talk will be based on current ongoing work with Sam Chow (Warwick) and Han Yu (Cambridge).

2019-20 Term 1

Organiser: Martin Orr

7 October

Daniel Gulotta (University of Oxford)

Vanishing theorems for Shimura varieties at unipotent level

We prove a vanishing result for the compactly supported cohomology of certain infinite level Shimura varieties. More specifically, if X_{K_p K^p} is a Shimura variety of Hodge type for a group G that becomes split over Q_p, and K_p is a unipotent subgroup of G(Q_p), then the compactly supported p-adic etale cohomology of X_{K_p K^p} vanishes above the middle degree.
We will also give an application to eliminating the nilpotent ideal in the construction of certain Galois representations.
This talk is based on joint work with Ana Caraiani and Christian Johansson and on joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.

14 October

Sam Chow (University of Warwick)

Rado's criterion over squares and higher powers

Given a finite colouring of the integers, is there a monochromatic Pythagorean triple? With Sofia Lindqvist and Sean Prendiville, we provide an affirmative answer in the analogous setting of generalised Pythagorean equations in five or more variables. Moreover, we show that a diagonal equation in sufficiently many variables has this property if and only if some non-empty subset of the coefficients sums to zero, which is a higher-degree version of Rado's characterisation of the linear case.
21 October

Christopher Daw (University of Reading)

Unlikely intersections in the moduli space of principally polarized abelian surfaces

The Zilber-Pink conjecture predicts that an irreducible algebraic curve V in $\mathcal{A}_2$, which is not contained in a proper special subvariety, has finite intersection with the special curves. The special curves in $\mathcal{A}_2$ comprise three families, namely, those curves parametrizing abelian surfaces with (1) quaternionic multiplication; (2) an isogeny to the square of an elliptic curve; (3) an isogeny to a product of elliptic curves, at least one of which has complex multiplication. In previous work, we handled the latter family assuming a large Galois orbits conjecture, and we established this conjecture when V satisfies certain conditions. In this talk, I will explain new work in which we handle the remaining families. The new ingredients are certain quantitative results in the reduction theory of algebraic groups.

28 October

Sarah Peluse (University of Oxford)

Bounds in the polynomial Szemerédi theorem

Let P_1,...,P_m be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemerédi’s theorem states that any subset A of {1,...,N} that contains no nontrivial progressions x,x+P_1(y),...,x+P_m(y) must satisfy |A|=o(N). In contrast to Szemerédi's theorem, quantitative bounds for Bergelson and Leibman's theorem (i.e., explicit bounds for this o(N) term) are not known except in very few special cases. In this talk, I will discuss recent progress on this problem.

4 November

Giada Grossi (UCL)

The p-part of BSD for residually reducible elliptic curves of rank one

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.

11 November

Chris Williams (University of Warwick)

p-adic L-functions for symplectic representations of GL(2n)

Let F be a totally real field, and let pi be an automorphic representation of GL(2n)/F that admits a Shalika model (that is, it is a transfer from GSpin(2n+1)). When pi is ordinary at p, recent independent work of Gehrman and Dimitrov--Januszewski--Raghuram gives a p-adic L-function attached to pi, that is, a p-adic measure interpolating its classical critical L-values. I will report on ongoing joint work with Daniel Barrera and Mladen Dimitrov where we generalise this to the non-ordinary case using overconvergent cohomology. Rather than standard overconvergent cohomology, defined with respect to the maximal torus in GL(2n), our results use a more flexible definition defined with respect to the subgroup GL(n) x GL(n), allowing weaker non-criticality conditions. I will start by giving a brief introduction to p-adic L-functions, and if time allows, will say a few words about our second main result, the variation of this construction in p-adic families.

18 November

Matthew Bisatt (University of Bristol)

Tame torsion of Jacobians and the inverse Galois problem

Fix a positive integer g and prime p. Does there exist a genus g curve, defined over the rationals, such that the mod p representation of its Jacobian is everywhere tamely ramified? We will give an affirmative answer to this question via the theory of hyperelliptic Mumford curves and give an application to a variant of the inverse Galois problem. This is joint work with Tim Dokchitser.

25 November

Kyle Pratt (University of Oxford)

Low-lying zeros of Dirichlet L-functions

I will present work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.

2 December

Pip Goodman (University of Bristol)

Restrictions on endomorphism algebras of Jacobians

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians of hyperelliptic curves C : y^2 = f(x) when the Galois group Gal(f) is insoluble and `large' relative to g the genus of C. But what happens when Gal(f) is not `large' or insoluble? We will see that for many values of g, much can be said if Gal(f) merely contains an element of `large' prime order.