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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

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)

9th March

Demi Allen (Bristol)

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.