Academic Year 2021/2022 - Number Theory Seminars

The Algebra and Number Theory group of the University of Luxembourg hosts three seminars. All of them are currently being held in a hybrid format.

Everyone is invited to attend! For more information, please contact Alexandre Maksoud, Andrea Conti or Gabor Wiese.


Luxembourg Number Theory Seminar

This seminar typically meets on Wednesdays at 13:30 or 14:00 in room MNO 1.030. You will find below a collection of abstracts.

Date (Room) Speaker Title
16/11/2021, 15:00 (online) Davide Lombardo Families of Jacobians with quaternion multiplication
24/11/2021, 14:00 (online) Alexandru Ciolan Cyclotomic numerical semigroups
08/12/2021, 15:00 (online) Jaclyn Lang tba
15/12/2021, 11:00-16:00 Emiliano Ambrosi, Riccardo Brasca, Christophe Cornut, Yukako KezukaLuxembourg Number Theory Day


Work in Progress Seminar

This seminar alternates with the Luxembourg Number Theory Seminar. Each session lasts 60 to 90 minutes. You will find below a collection of abstracts.

Date (Room) Speaker Title
13/10/2021, 14:00 (MNO 1.030) Sebastiano Tronto A generalization of injective modules
20/10/2021, 14:00 (MNO 1.030) Alexandre Maksoud The Gross-Kuz'min conjecture via transcendance theory
27/10/2021, 14:00 (MNO 1.030) Nikita Karpenko Indexes of grassmannians for spin groups
3/11/2021, 14:00 (MNO 1.030) Sebastiano Tronto A category of division modules
10/11/2021, 14:00 (MNO 1.030) Andrea Conti Galois representations attached to eigenforms of infinite slope
01/12/2021, 14:00 (MNO 1.030) Fabio La Rosa On a question of Serre (joint work with C. Khare and G. Wiese)


Working Group: seminar on perverse sheaves (Winter semester 2021)

The seminar will take place on Thrusdays from 14:00 to 15:30 in room MNO 1.010. Here is the program of the seminar. Recordings are available upon request to the organizers.

Date (Room) Speaker Title
11/10/2021, 14:00 (MNO 1.030) Alfio Fabio La Rosa Overview
21/10/2021, 14:00 (MNO 1.050) Sebastiano Tronto Étale cohomology I: Construction of the étale site and definition of the étale cohomology of a sheaf
28/10/2021, 14:00 (MNO 1.020) Bryan Advocaat Étale cohomology II: some computations
04/11/2021, 14:00 (MNO 1.020) Andrea Conti Étale cohomology IIIa: l-adic sheaves and Grothendieck's trace formula
Étale cohomology IIIb: Frobenius actions on cohomology
11/11/2021, 14:00 (MNO 1.010) Alisa Govzmann Derived Categories I: t-structures on triangulated categories and their hearts
18/11/2021, 14:00 (MNO 1.010) Flavio Perissinotto Derived Categories II: Cohomological functors
25/11/2021, 14:00 (MNO 1.010) Alexandre Maksoud Perverse sheaves I
02/12/2021, 14:00 (MNO 1.010) Swann Tubach Perverse sheaves II
09/12/2021, 14:00 (MNO 1.010) ?? The l-adic Fourier transform
16/12/2021, 14:00 (MNO 1.010) ?? Gabber's Decomposition Theorem

Outline

The Weil Conjectures are a set of four statements concerning the zeta-function of a smooth projective variety over a finite field:

These conjectures were proved by Deligne using the formalism of étale cohomology: three of them follow directly from the properties of this cohomology theory, while the Riemann Hypothesis is harder.

Laumon gave an elegant proof of the Riemann Hypothesis using the theory of Perverse Sheaves. This is one of many striking applications that the theory found in various areas of Mathematics, from the representation theory of Lie algebras to enumerative and arithmetic geometry. Recently, it has been applied by Scholze and Fargues in their work on the geometrisation of the Local Langlands Conjectures.

The idea of this seminar is to introduce just enough l-adic étale cohomology theory in order to better appreciate how the Weil conjectures, minus the Riemann Hypothesis, follow from its existence. We will subsequently develop the theory of Perverse Sheaves and understand how this is used to prove the Riemann Hypothesis. See the program for a more detailed plan.

Webex information

Meeting number: 2733 415 1759 Pwd: MspGzFi3r93


Collection of abstracts

Sebastiano Tronto (uni.lu) A generalization of injective modules

The underlying abelian group of the field of rational numbers Q has an interesting property: it is divisible, which means that for every element x of Q and every positive integer n there is an element y of Q such that ny = x. On the other hand, if we only care about dividing by the powers of a certain prime, then also the underlying abelian group of the ring Z[p^{-1}] has a similar property: it is p-divisible, that is for every element x there is an element y such that py = x. If one tries to generalize these concepts to modules over a general (associative, unitary) ring R, things may not work so well, among other things due to the possible presence of zero-divisors in the base ring. There is however a natural (or categorical) concept that works well over any ring, which is injectivity. Indeed an abelian group is injective as a Z-module if and only if it is divisible. What is in this setting a suitable generalization for p-divisibility? Is there a more general property that includes divisibility and p-divisibility as special cases, and that also works well for R-modules? In this talk I will propose a definition that provides a positive answer to the two questions above. If time permits I will also show an analogue of Morita duality using this more general definition.

Alexandre Maksoud (uni.lu) The Gross-Kuz'min conjecture via transcendance theory

The Gross-Kuz'min conjecture deals with Galois descent of class groups in the context of Zp-extensions of number fields. It is closely related to Leopoldt's conjecture and has many applications to classical Iwasawa theory. The aim of this talk is to state and deduce new cases of this conjecture from results of Waldschmidt and Roy in p-adic transcendence theory.

Nikita Karpenko (University of Alberta) Indexes of grassmannians for spin groups

Based on a recent joint work with Rostislav Devyatov and Alexander Merkurjev, the sharp upper bounds on indexes of twisted grassmannians for the spin groups Spin(n) are being established. Equivalently, for every integer m in the interval [1,n/2], we are looking for the greatest common divisor of degrees of the finite base field extensions over which the Witt index of a generic n-dimensional quadratic form of trivial discriminant and Clifford invariant becomes at least m. The case of m=[n/2] was done by Burt Totaro in 2005.

Sebastiano Tronto (uni.lu) A category of division modules

Let G be a commutative algebraic group over a field K of characteristic zero. We are interested in studying the smallest field extension of K that contains the coordinates of all the points of G over some algebraic closure of K that have a multiple in G(K), or other similar field extensions. In order to do so we first need to understand certain properties of G(K) as a module over the ring of K-endomorphisms of G, and in particular its "division extensions". Using the theory of J-injective modules introduced in my previous talk we will construct a category that in a sense describes all such extensions.

Andrea Conti (uni.lu) Galois representations attached to eigenforms of infinite slope

Classical eigenforms of finite p-slope are p-adically interpolated by the rigid analytic eigencurve, an object that admits a precise interpretation in terms of a Galois deformation space. No such construction is available for eigenforms of infinite p-slope. We use techniques from Galois deformation theory to study whether the representations attached to such forms can be p-adically interpolated, with the goal to show that this is possible only in some exceptional cases, and to deduce an analoguous result for the eigenforms themselves.

Davide Lombardo (Università di Pisa) Families of Jacobians with quaternion multiplication

In joint work with Victoria Cantoral-Farfán and John Voight we investigate families of even-dimensional Jacobians defined over Q and admitting an action of the quaternion group. Such abelian varieties are unusual in several ways: for example, their ring of algebraic cycles is not generated by divisor classes, a fact which has consequences both on their arithmetic and on their geometry. We prove that -- for every even dimension greater than two -- 100% of the members of the families we consider satisfy the Hodge, Tate and Mumford-Tate conjectures, and provide explicit generators for their Hodge rings. As a consequence, we show that for such abelian varieties A the minimal field of definition of the endomorphisms and the minimal field over which the Galois representations attached to A have connected image are different.

Alexandru Ciolan (MPIM Bonn) Cyclotomic numerical semigroups

In this talk I will speak about the relatively new concept of cyclotomic numerical semigroups, that is, numerical semigroups S whose associated semigroup polynomials P_S(x)=(1-x)\sum_{s \in S}x^s factorize into cyclotomic polynomials. In particular, I will give a summary of the tools we currently have at our disposal for studying connections between cyclotomic polynomials and numerical semigroups, such as Betti elements, cyclotomic exponent sequences, depths and heights. I will also report on recent developments made towards solving a conjecture formulated by the speaker, García-Sánchez and Moree (2016), which says that a numerical semigroup is cyclotomic if and only if it is a complete intersection, and I will present an interesting application of cyclotomic numerical semigroups to elliptic curve cryptography. This is joint work with Pedro García-Sánchez, András Herrera-Poyatos and Pieter Moree.

Fabio La Rosa (uni.lu) On a question of Serre (joint work with C. Khare and G. Wiese)

J. P. Serre, in the article "On a theorem of Jordan", considers a certain family of polynomials with integer coefficients. For the polynomials of degree up to 4, he is able to find appropriate automorphic objects that encode the number of roots of the reduction of the polynomials modulo p. To treat the degree 5 case in an analogous fashion, following Calegari we establish a modularity result for the standard representation of certain Galois extensions of the rational numbers which realise the symmetric group on 5 letters as Galois group. This allows us to relate the Hecke eigenvalue at p of 4-dimensional automorphic representation over the rational numbers to the number of roots of the reduction modulo p of the polynomial.


Last modification: 25 November 2021.