Skip to main content

Number Theory Seminar

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

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