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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

2018-19 Term 3

Organiser: Martin Orr

29 April

Fabien Pazuki (University of Copenhagen)

Regulators of number fields and abelian varieties

In the general study of regulators, we present three inequalities. We first bound from below the regulators of number fields, following previous works of Silverman and Friedman. We then bound from below the regulators of Mordell-Weil groups of abelian varieties defined over a number field, assuming a conjecture of Lang and Silverman. Finally we explain how to prove an unconditional statement for elliptic curves of rank at least 4. This third inequality is joint work with Pascal Autissier and Marc Hindry.

7 May

Tuesday 3pm MS.03

Nicole Looper (University of Cambridge)

Dynamical uniform boundedness and the abc-conjecture

In 1994, Morton and Silverman formulated what is now known as the Uniform Boundedness Conjecture. A major open problem in arithmetic dynamics, the conjecture includes as particular cases the theorems of Mazur and Merel on uniform bounds on the rational torsion points on elliptic curves. In this talk, I will discuss a uniform boundedness theorem for unicritical polynomials. I will also discuss the relevant tools from Diophantine geometry and their relation to the abc-conjecture.

13 May

Kevin Ford (University of Illinois at Urbana-Champaign)

Prime number models, large gaps, prime tuples and the square-root sieve

We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

20 May

Kirsti Biggs (University of Bristol)

Heights of algebraic numbers and Lehmer's conjecture

In 1933, D.H. Lehmer asked a question about roots of integer polynomials which can be translated into the language of the Weil height of algebraic numbers. Although his question - now known as Lehmer's conjecture - remains open, it is possible to prove bounds of the required type in certain specific cases. In this talk, I'll discuss a generalisation of a result of Amoroso and Masser to algebraic numbers generating Galois extensions of an arbitrary number field, which is joint work with S. Akhtari, K. Aktas, A. Hamieh, K. Petersen and L. Thompson.

27 May No seminar
3 June

Alice Pozzi (UCL)

The eigencurve at Eisenstein weight one points

In this talk, we discuss the geometry of the Coleman-Mazur eigencurve at weight one Eisenstein points. The local nature of the eigencurve is mostly understood at classical points of weight greater than one, while one observes some unusual behaviours at weight one. In particular, we study cuspidal Hida families specializing to Eisenstein series at weight one. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations.

We discuss the implications of our analysis on the classicality of a certain overconvergent eigenspace. Finally, we explain how this Galois-theoretic method yields some new insight on Gross’s formula relating the leading term of the p-adic L-function to a Stark unit. This is joint work with Adel Betina and Mladen Dimitrov.

10 June

Maxim Gerspach (ETH Zürich)

Even moments of random matrix-valued multiplicative functions

Random multiplicative functions have received a lot of attention recently as random models for number-theoretic objects such as Dirichlet characters or the Möbius function, and understanding their behaviour has allowed us to get heuristic, and sometimes even rigorous, insights into the behaviour of their deterministic counter-parts.

I will introduce a generalization of random multiplicative functions which allows matrix values, and I will give some bounds on their moments. These turn out to be related to the so-called generalized joint spectral radii, which have been studied in a wide range of fields such as dynamical systems, wavelets and control theory.

17 June

Alessandro Arlandini (Warwick)

Factorisation of the p-adic Rankin-Selberg L-function in the supersingular case

In this talk I will discuss how to express the p-adic Rankin-Selberg L-function as a product of other p-adic L-functions. In order to do this I will rely on the existence of Euler systems, in particular of the Beilinson-Flach one. The crucial point of the strategy is the relation between the base class of an Euler system and some special value of the attached complex and p-adic L-functions. I will focus on the case of a supersingular modular form, which extends results already known in the ordinary case.

24 June

Adam Morgan (University of Glasgow)

Parity of Selmer ranks in quadratic twist families

We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun–Mazur–Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.

2018-19 Term 2

Organiser: Martin Orr

7 January No seminar
14 January No seminar
21 January

Kevin Hughes (University of Bristol)

Discrete restriction theory

We will introduce “discrete restriction theory” and it’s applications in number theory and analysis. We will then discuss Trevor Wooley’s efficient congruencing method and a prove “discrete decoupling” result for the parabola. If time permits I will discuss new bounds for discrete restriction to the curve (x,x^3). This is an example which presently lies beyond the scope of efficient congruencing/decoupling. This is joint work with Trevor Wooley.

28 January

Johannes Sprang (Universität Regensburg)

Eisenstein–Kronecker series via the Poincaré bundle

A classical construction of Katz gives a purely algebraic construction of real-analytic Eisenstein series using the Gauss–Manin connection on the universal elliptic curve. This has many applications in number theory. We provide an alternative algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle. This construction allows a new interpretation of Katz’ p-adic Eisenstein measure in terms of p-adic theta functions. If time permits, we will discuss applications to the study of the elliptic polylogarithm for families of elliptic curves.

4 February

Jack Shotton (Durham University)

Shimura curves and Ihara's lemma

Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was the key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves.

11 February

Samuel Le Fourn (University of Warwick)

Runge and Baker-type methods for integral points in higher dimension

I will present a new result of explicit finiteness of integral points on some quasi-projective varieties, drawing inspiration from both Runge's method and Baker's method (well-known in the case of curves). I will spend most of the talk explaining the main ideas of the proof, and how one can adapt it to various situations thanks to the fundamental simplicity of the latter, without forgetting explicit examples.

18 February

Ariel Pacetti (Universidad de Cordoba)

On the number of Galois orbits of newforms

In this talk we will present a lower bound for the number of Galois orbits of newforms for $S_k(\Gamma_0(N))$ for $k$ big enough, in terms of some arithmetic invariants. This is a joint work with Luis Dieulefait and Panagiotis Tsaknias.

25 February

Rodolphe Richard (University of Cambridge)

Toward an 'arithmetic' variant of André-Oort conjecture

We present a non trivially false arithmetic generalisation of André-Oort conjecture. Indeed we prove it in two non trivial cases (one, under GRH is j./w. Edixhoven). We relate it, and motivate by, recent trends in equidistribution.

4 March

Catherine Hsu (University of Bristol)

Higher Eisenstein Congruences

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we re-examine Eisenstein congruences, incorporating a notion of "depth of congruence," in order to understand the local structure of Eisenstein ideals associated to weight 2 cusp forms of squarefree level.

11 March

Francesca Bianchi (University of Oxford)

Extra points in Chabauty-type methods

Let $G$ be the set of rational points on a smooth projective curve of genus at least $2$ or the set of integral points on an elliptic curve over $\mathbb{Q}$. When applicable, the Chabauty–Coleman–Kim methods identify $G$ as a subset of a finite set of $\mathbb{Q}_p$-rational points $L$. But what points can arise in $L\setminus G$? We discuss this question in a Chabauty–Coleman and a Chabauty–Kim setting in which $L$ is defined by two $p$-adic equations; this is connected with conjectures of Stoll and Kim.
Part of the talk is joint work with Jennifer Balakrishnan, Victoria Cantoral-Farfán, Mirela Çiperiani and Anastassia Etropolski.

2018-19 Term 1

Organiser: David Lowry-Duda

8 October

Samir Siksek (University of Warwick)

On the asymptotic Fermat conjecture

15 October

Ariel Weiss (University of Sheffield)

Irreducibility of Galois representations associated to low weight Siegel modular forms

22 October

Kim Logan (University if Minnesota)

Zeros of $L$-functions and unbounded operators

29 October

Martin Orr (University of Warwick)

Unlikely intersections and E x CM abelian surfaces

5 November

Olivia Beckwith (University of Bristol)

Indivisibility and divisibility of class numbers of imaginary quadratic fields

12 November

Thomas Bloom (Cambridge University)

Diophantine approximation, GCD sums, and the sum-product phenomenon

19 November

Alain Kraus (Université Pierre-et-Marie-Curie - Paris VI)

Asymptotic Fermat's Last Theorem and cyclotomic Z_2-extensions

26 November

Kwok-Wing Tsoi (King's College)

On higher special elements of p-adic representations

3 December

Nuno Freitas (University of Warwick)

The modular method, Frey abelian varieties and Fermat-type equations

7 December (Friday)

Minhyong Kim (Oxford)

Diophantine geometry and principal bundles