London Number Theory Seminar

The London Number Theory Seminar is held weekly, on Wednesdays, during term time. The location of the seminar cycles between KCL, Imperial College and UCL.

This term (Autumn 2015), the seminar will be hosted by Imperial College, and will be held on Wednesdays from 4pm to 5pm in room 140 in the Huxley Building. The seminar is organized by David Helm, and starts on 7th October.

The talks will be preceded by tea/coffee in Imperial's common room (room 549 Huxley) from around 3:30.

07/10/15 Martin Orr (Imperial)
Title: A bound for rational representations of isogenies in the fundamental set
Abstract: Let z and z' be two points in the standard fundamental set in the upper half-plane. If the corresponding elliptic curves are related by an isogeny of degree N, then there is a 2x2 matrix with integer coefficients and determinant N which maps z to z'. As an ingredient in their work on unlikely intersections, Habegger and Pila proved that the entries of this matrix are bounded by a uniform polynomial in N. I will discuss the generalisation of this result to moduli of abelian varieties and beyond, to Riemannian symmetric spaces of non-compact type.

14/10/15 Stephane Bijakowski (Imperial)
Title: The partial degrees of the canonical subgroup
Abstract: If the Hasse invariant of a p-divisible group is small enough, then one can construct a canonical subgroup inside its p-torsion. I will first present an alternative approach to this problem, assuming the existence of a subgroup satisfying some simple conditions. A key property is the relation between the Hasse invariant and the degree of the canonical subgroup. When one considers a p-divisible group with extra structures, more information is available. I will define the partial Hasse invariants, the partial degrees, and relate them for the canonical subgroup.

21/10/15 Yiwen Ding (Imperial)
Title: L-invariants and local-global compatibility for GL2
Abstract: Let F be a totally real number field, w a prime of F above p, V a 2-dimensional p-adic representation of the absolute Galois group G_F of F which appears in etale cohomology of quaternion Shimura curves. When the restriction Vw of V to the decomposition group of G_F at w is semi-stable non-crystalline in Fontaine's sense, we can associate to Vw the so-called Fontaine-Mazur L-invariants, which are invisible in classical local Langlands correspondance. We show that these L-invariants can be found in the completed cohomology group of Shimura curves.

28/10/15 Rob Kurinczuk (Bristol)
Title: Cuspidal l-modular representations of classical p-adic groups
Abstract: For a classical groups (unitary, special orthogonal, symplectic) over locally compact non-archimedean fields of odd residual characteristic p, Shaun Stevens has developed an approach to studying their (smooth) complex representations based on the theory of types of Bushnell and Kutzko. I will describe some joint work with Shaun Stevens, in which we relate positive level cuspidal representations in Stevens' construction to level zero cuspidal representations in certain associated groups and consider a generalisation to modular representations in characteristic prime to p.

04/11/15 Paul Ziegler
Title: p-kernels occurring in isogeny classes of p-divisible groups
Abstract: I will give a criterion which allows to determine, in terms of the combinatorics of the root system A_n, which p-kernels occur in a given isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. This question is related to the relationship between Newton and Ekedahl-Oort strata on reductions of Shimura varieties as well as the non-emptiness of affine Deligne-Lusztig varieties.

9/11/15 London-Paris Number Theory Seminar (in Paris)

11/11/15 Laurent Berger
Title: Iterated extensions and relative Lubin-Tate groups
Abstract: An important construction in p-adic Hodge theory is the 'field of norms' corresponding to an infinite extension K_infty/K. For the cyclotomic extension, it is possible to lift the field of norms to characteristic zero, and we can ask for which other extensions K_infty/K this is possible. The goal of this talk is to explain this question and discuss some partial answers. This involves p-adic dynamical systems, Coleman power series and relative Lubin-Tate groups.

18/11/15 Julien Hauseux (Kings)
Title: Parabolic induction and extensions
Abstract: Let G be a p-adic reductive group. We describe the extensions between admissible smooth mod p representations of G which are parabolically induced from supersingular representations of Levi subgroups. More precisely, we determine which extensions do not come from parabolic induction. In order to do so, we partially compute Emerton's delta-functor of derived ordinary parts on any parabolically induced representation of G. These computations work with mod p^n coefficients, thus some of the results on extensions can be lifted in characteristic 0 for admissible unitary continuous p-adic representations of G.

25/11/15 Arne Smeets (Imperial)
Title: Logarithmic good reduction and cohomological tameness
Abstract: I will discuss two notions of tameness for varieties defined over a field equipped with a discrete valuation, which are only interesting if the residual characteristic is positive: cohomological tameness, and logarithmic good reduction. The first notion is weaker than the second one (Nakayama). I will explain why these notions are equivalent in the case of abelian varieties; this can be seen as a logarithmic version of the Néron-Ogg-Shafarevich criterion (joint work with A. Bellardini). I will also discuss a cohomological trace formula for the tame monodromy operator, conjectured by Nicaise for cohomologically tame varieties, proven by the speaker for varieties with logarithmic good reduction.

2/12/15 Jacques Benatar (Kings)
Title: TBA

9/12/15 Jennifer Balakrishnan (Oxford)
Title: TBA

16/12/15 Daniel Skodlerack (Imperial)
Title: TBA

The seminar will be preceded by the Study Groups, which this term will run 1200-1330 (p-adic Langlands) and 1400-1530 (the yoga of weights).

A list of previous seminar talks is here.

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