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. |
2018-19 Term 1
Organiser: David Lowry-Duda
8 October |
Samir Siksek (University of Warwick) |
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 |