London Number Theory Seminar

UCL King's Imperial College


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 (Summer 2018), the seminar will be hosted by Kings College, and will be held on Wednesdays in KCL room K3.11, starting at 1600 (unless specified otherwise). The organisers are Nadav Yesha and James Newton.

The talks start on Wed 25th April and finish on 27th June. The talks will be preceded by tea at 1530 in S5.21.

Important note: The organisers maintain their own seminar website here, which might well be more up to date than this one when it comes to titles and abstracts.

25/04/18 Peter Humphries (UCL)
Title: Quantum unique ergodicity in almost every shrinking ball
Abstract: I will discuss the problem of small scale equidistribution of Hecke-Maass eigenforms, namely the problem of the rate at which hyperbolic balls can shrink as the Laplacian eigenvalue tends to infinity for which the Laplacian eigenfunctions still equidistribute on these balls. There is a natural barrier - the Planck scale - for which equidistribution fails, but conditionally equidistribution occurs in almost every shrinking ball at every larger scale. I will also discuss related small scale equidistribution problems for geometric invariants associated to quadratic fields.

2/5/18 Alexandra Florea (Bristol)
Title: Moments of cubic L-functions over function fields
Abstract: I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula with a (maybe a little surprising) main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields.

9/5/18: This week the seminar will take place at 1400 (in K3.11), to allow people to attend Sarah Zerbes' inaugural lecture at UCL.
Steve Lester (QMUL)
Title: Sign changes of Fourier coefficients of half-integral weight modular forms.
Abstract: For a square-free integer n, Waldspurger showed that square of the nth Fourier coefficient of a half-integral weight Hecke cusp form is proportional to the central value of an L-function. It remains to understand the sign of the coefficient. In this talk I will discuss joint work with Maks Radziwill (McGill) in which we study the number of sign changes of coefficients of such forms.

16/5/18 Holly Krieger (Cambridge)
Title: Title: A dynamical approach to common torsion points
Abstract: Bogomolov-Fu-Tschinkel conjectured that there is a uniform upper bound on the number of common torsion points of two nonisomorphic elliptic curves (more precisely, on the number of common images of torsion points when the curve is presented as a double cover of the Riemann sphere). This is an example of the phenomenon of unlikely intersections in arithmetic geometry. I will discuss a dynamical approach to this conjecture via Lattès maps of the Riemann sphere associated to an elliptic curve. I will report on recent progress on this dynamical approach (joint with Laura DeMarco and Hexi Ye) and formulate a more general dynamical conjecture.

23/5/18 Jan Nekovář (Paris)
Title: Semisimplicity of certain Galois representations occurring in étale cohomology of unitary Shimura varieties
Abstract: Conjecturally, the category of pure motives over a finitely generated field $k$ should be semisimple. Consequently, $\ell$-adic étale cohomology of a smooth projective variety over $k$ should be a semisimple representation of the absolute Galois group of $k$. This was proved by Faltings for $H^1$, as a consequence of his proof of Tate's conjecture. In this talk, which is based on a joint work with K. Fayad, I am going to explain a proof of the semisimplicity of the Galois action on a certain part of étale cohomology of unitary Shimura varieties. The most satisfactory result is obtained for unitary groups of signature $(n,0)^a \times (n-1,1)^b \times (1,n-1)^c \times (0,n)^d$.

29/5/18 to 30/5/18 -- the London-Paris Number Theory Seminar.

30/5/18 This week the seminar (and algebraic study group) will take place in room 505 at UCL, as the London–Paris seminar is at UCL in the morning.
Matthew Morrow (Paris)
Title: TBA
Abstract: TBA

6/6/18 Arno Kret (Amsterdam)
Title: Galois representations for the general symplectic group.
Abstract: In a recent preprint with Sug Woo Shin arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain this result and some parts of the construction.

13/6/18 Chris Birkbeck (UCL)
Title: Slopes of Hilbert modular forms near the boundary of weight space.
Abstract: Recent work of Liu–Wan–Xiao has proven in many cases how slopes of modular forms behave near the boundary of weight space, giving us insights into the geometry of the associated eigenvarieties. One can ask if there is similar behaviour in the case of Hilbert modular forms. I will discuss some conjectures on how the slopes should behave near the boundary as well as explaining why the methods of Liu–Wan–Xiao do not appear to extend to the Hilbert case. Lastly, I will discuss some recent examples where it is possible to partially prove these conjectures in the case when chosen prime is inert.

20/6/18 Djordjo Milovic (UCL)
Title: Spins of ideals and arithmetic applications to one-prime-parameter families
Abstract: We will define three similar but different notions of "spin" of an ideal in a number field, and we will show how a number-field version of Vinogradov's method (a sieve involving "sums of type I" and "sums of type II") can be used to prove that spins of prime ideals oscillate. Such equidistribution results have applications to the distribution of 2-parts of class groups of quadratic number fields in thin families parametrized by prime numbers. Parts of this talk are joint work with Peter Koymans.

27/6/18 Kęstutis Česnavičius (Paris)
Title: Purity for the Brauer group
Abstract: A purity conjecture due to Grothendieck and Auslander--Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension \ge 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0, p). We will discuss an approach to the mixed characteristic case via the tilting equivalence for perfectoid rings.

The seminar will be preceded by various study groups.

A list of previous seminar talks is here.


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