London Number Theory Seminar
King's College of London

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London Number Theory
Seminar
(Summer 2019)

Contact:

Department of Mathematics,
King's College of London,
Strand, London,
WC2R 2LS

eran.assaf@kcl.ac.uk


London Number Theory Seminar

During the summer term of 2019, James Newton and I are organizing the London Number Theory Seminar hosted at KCL.


Important Notice: On July 3rd, 2019, the seminar will be held in S4.23 (Strand building, 4th floor) and not in the usual place.


The seminar is held on Wednesdays at 4pm in the Mathematics Department (Strand Campus, Strand, WC2R 2LS) Room K-1.56 (King's building, Floor (-1)).

As usual, before the seminar there will be tea and coffee available in the common room on the 5th floor. Earlier in the afternoon there is an algebraic study group based on the paper "On the generic part of the cohomology of compact unitary Shimura varieties" held in Room S4.23 (Strand building, Floor 4).

For more information you can sign up to one of the London mailing lists available here, where you can also find information about previous talks.

The updated list of speakers and dates is as follows.


Schedule:

  • 24 Apr 2019 - Jesse Jääsaari (University of Helsinky)

    • Title: Exponential Sums Involving Fourier Coefficients of higher rank automorphic forms

    • Abstract: In this talk I will describe various conjectures concerning correlations between Fourier coefficients of higher rank automorphic forms and different exponential phases. I will also discuss recent work (partly in progress) towards some of these conjectures.


  • 01 May 2019 - Kazim Büyükboduk (UC Dublin)

    • Title: Rank-2 Euler systems for non-ordinary symmetric squares

    • Abstract: According to Beilinson's conjectures, the leading term of an L-function at a "generic" non-critical point should be explained by special elements in the highest exterior power of the relevant motivic cohomology. p-adic (étale) realisations of these special elements would in turn give rise to (higher rank) Euler systems. I will report on joint work with A. Lei, where we obtain a rank-2 Euler system associated to the symmetric square of an eigenform at a non-ordinary prime p twisted by a Dirichlet character, whose non-triviality is accounted by p-adic L-values.


  • 07-08 May 2019 - London-Paris Number Theory Seminar

  • 08 May 2019 - Ben Heuer (King's College London)

    • Title: perfectoid modular forms and a tilting isomorphism at the boundary of weight space

    • Abstract: Similarly to how complex modular forms are defined as functions on the complex upper half plane, Chojecki--Hansen--Johansson describe p-adic modular forms as functions on Scholze's perfectoid modular curve at infinite level. In this talk, we show that the appearance of perfectoid spaces in this context is not just a technical coincidence, but that this definition gives rise to 'perfectoid phenomena' appearing in the world of p-adic and classical modular forms. As an example of this, we discuss a tilting isomorphism of p-adic modular forms near the boundary of weight space which gives a new perspective on the space of T-adic modular forms defined by Andreatta--Iovita--Pilloni. This isomorphism can be explained by a theory of 'perfectoid modular forms' that we will also discuss in this talk.


  • 15 May 2019 - [Cancelled, due to workshop this week - The p-adic Langlands Programme and Related Topics]

  • 22 May 2019 - Eva Viehmann (Technical University of Munich)

    • Title: Affine Deligne-Lustig varieties

    • Abstract: Affine Deligne-Lusztig varieties are defined as certain subschemes of affine flag varieties using Frobenius-linear algebra. They are used in arithmetic geometry to describe the reduction of Shimura varieties. Motivated by this relation, I will report on recent geometric results describing affine Deligne-Lusztig varieties, and applications.


  • 29 May 2019 - Eugenia Rosu (University of Arizona)

    • Title: Special cycles on orthogonal Shimura varieties

    • Abstract: Extending on the work of Kudla-Millson and Yuan-Zhang-Zhang, together with Yott we are constructing special cycles for a specific GSpin Shimura variety. We further construct a generating series that has as coefficients the cohomology classes corresponding to the special cycle classes on the GSpin Shimura variety and show the modularity of the generating series in the cohomology group over C.


  • 05 June 2019 - Paul Ziegler (University of Oxford)

    • Title: Geometric stabilization via p-adic integration

    • Abstract: The fundamental lemma is an identity of integrals playing an important role in the Langlands program. This identity was reformulated into a statement about the cohomology of moduli spaces of Higgs bundles, called the geometric stabilization theorem, and proved in this form by NgĂ´. I will give an introduction to these results and explain a new proof of the geometric stabilization theorem, which is joint work with Michael Groechenig and Dimitri Wyss, using the technique of p-adic integration.


  • 12 June 2019 - Ramla Abdellatif (Université de Picardie Jules Verne)

    • Title: Restricting p-modular representations of p-adic groups to minimal parabolic subgroups

    • Abstract: Given a prime integer $p$, a non-archimedean local field $F$ of residual characteristic $p$ and a standard Borel subgroup $P$ of $GL_{2}(F)$, Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas proved that the restriction to $P$ of (irreducible) smooth representations of $GL_{2}(F)$ over $\overline{\mathbb{F}}_{p}$ encodes a lot of information about the full representation of $GL_{2}(F)$ and that it may leads to useful statement about $p$-adic representations of $GL_{2}(F)$. Nevertheless, the methods used by Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas at that time heavily rely on the understanding of the action of certain spherical Hecke operator and on some combinatorics specific to the $GL_{2}(F)$ case. This can be carried to other specific quasi-split groups of rank $1$, but this is not very satisfying.\\ In this talk, I will report on a joint work with J. Hauseux. Using an different approach based on Emerton's ordinary parts functor, we get a more uniform context which shed a new light on Pa${\check{\text{s}}}$k$\overline{\text{u}}$nas' results and allows us to get a natural generalization of these results for arbitrary rank $1$ groups. In particular, we prove that for such groups, the restriction of supersingular representations to a minimal parabolic subgroup is always irreducible.


  • 19 June 2019 - Mikhail Gabdullin (Lomonosov Moscow State University)

    • Title: On the stochasticity parameter of quadratic residues

    • Abstract: Let $U=\{ 0 \le u_1 < u_2 < ... < u_k < M \}$ be arbitrary subset of residues modulo $M$; set also $u_{k+1}:=M+u_1$. V.I.Arnold defined the stochasticity parameter of the set $U$ to be the quantity $\sum_{i=1}^k (u_{i+1}-u_i)^2$ (the sum of squares of the distances between elements of $U$), and it turns out that too small or too large values of $S(U)$ indicate that $U$ is far from a random set: for a fixed $k$, $S(U)$ is minimal when the points of $U$ are equiditributed and $S(U)$ is maximal when $U$ is an interval. M.Z.Garaev, S.V.Konyagin and Yu.V.Malykhin studied the stochasticity parameter of quadratic residues modulo a prime and showed that it is asymptotically equal to the stochasticity parameter of a random set of the same size. We turn to this problem arbitrary modulo $M$ and prove the same asymptotics for a set of moduli of positive lower density; we are also able to show that for these moduli the parameter of quadratic residues is in fact less than the parameter of a random set of the same size. Also we will discuss how (potentially) this result can be extended for almost all moduli.


  • 26 June 2019 - Daniel Gulotta (University of Oxford)

    • Title: Vanishing theorems for Shimura varieties at unipotent level

    • Abstract: We prove a vanishing result for the compactly supported cohomology of certain infinite level Shimura varieties. More specifically, if $X_{K_pK^p}$ 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.


  • 3 July 2019 - Christopher Frei (University of Manchester)

    • Title: Average bounds for l-torsion in class groups.

    • Abstract: Let $l$ be a positive integer. We discuss average bounds for the $l$-torsion of the class group for some families of number fields, including degree-$d$-fields for $d$ between $2$ and $5$. Refinements of a strategy due to Ellenberg, Pierce and Wood lead to significantly improved upper bounds on average. The case $d=2$ implies the currently best known upper bounds for the number of $D_p$ - fields of bounded discriminant, for odd primes $p$. This is joint work with Martin Widmer. (The results presented here are different from those presented by Martin Widmer in his talk with a similar title in Jan 2018.)