Academic Year 2024/2025 - Number Theory Seminars and Courses

The Algebra and Number Theory group of the University of Luxembourg hosts two seminars and some courses.

• Seminars
  • The Luxembourg Number Theory Seminar hosts invited speakers.
  • The Work in Progress Seminar gives the opportunity to members of the group to present their work in progress. It alternates with the Luxembourg Number Theory Seminar.
• Courses
  • Course/workshop on Kummer Theory, by Antonella Perucca from the University of Luxembourg (September 9 to 12)
  • Course on Abelian Varieties and Isogeny Graphs, by Jean Kieffer from the Université de Lorraine (November 14, 15, 28 and 29)

Everyone is invited to attend! For more information, contact the organiser Félix Baril Boudreau.

Luxembourg Number Theory Seminar (Fall 2024)

Work in Progress Seminar

Course on Abelian Varieties and Isogeny Graphs, by Jean Kieffer (Université de Lorraine)

* Review of abelian varieties, polarizations, Jacobians

Reference: Milne's course notes "Abelian varieties"

* Isogeny classes over finite fields, Honda-Tate theory; ordinary and supersingular classes

Reference: Waterhouse--Milne, "Abelian varieties over finite fields"

* Classification of abelian varieties within an isogeny class

Reference: Waterhouse, "Abelian varieties over finite fields"

* The zoology of isogeny graphs

References: Kohel, "Endomorphism rings of elliptic curves over finite fields"; Brooks--Jetchev--Wesolowski, "Isogeny graphs of ordinary abelian varieties"

Sequence 2: Topics in Isogeny-based cryptography

* Hard problems in isogeny graphs

References: various cryptography papers

* Equidistribution in isogeny graphs

References: Pizer, "Ramanujan graphs and Hecke operators"; Jao--Miller--Venkatesan, "Expander graphs based on GRH with an application to elliptic curve cryptography"

* Computing isogenies

References: various mathematical papers

* Embedding isogenies

References: Robert, "Breaking SIDH in polynomial time"; Robert, "Some applications of higher-dimensional isogenies to elliptic curves"

Course/workshop on Kummer Theory, by Antonella Perucca (uni.lu)

Abstracts for the Luxembourg Number Theory Seminar

Francisco García Cortés (Universidad de Sevilla) Finite monodromy for some Belyi related two-parameter exponential sums (16/10/2024)

Original results are joint work with Antonio Rojas-León. Let \(\mathbb{F}_q\) be a finite field of characteristic \(p\), \(\ell\neq p\) a prime, \(\psi:\mathbb{F}_q\to\mathbb{C}^\times\sim\overline{\mathbb{Q}}_\ell^\times\) a non-trivial additive character and \(f(x)\in\mathbb{F}_q[x]\) a polynomial not Artin--Schreier equivalent to a linear polynomial.

Consider the complex of \(\ell\)-adic sheaves on \(\mathbb{G}_{m,\mathbb{F}_q}\times\mathbb{G}_{m,\mathbb{F}_q}\) whose trace function at \((s,t)\in \mathbb{F}_{q^r}^\times\times\mathbb{F}_{q^r}^\times\) is \[-\frac{1}{q^{r/2}}\sum_{x\in\mathbb{F}_{q^r}} (\psi\circ\mathrm{Tr}_{\mathbb{F}_{q^r}/\mathbb{F}_q})(sf(x)+tx).\] Up to restriction to a certain dense open subset and shifting, this is a local system \(\mathcal{F}_f\). That is, we can speak about its monodromy.

Our main question is: For which polynomials \(f(x)\) does \(\mathcal{F}_f\) have a finite monodromy group?

We start by recalling the recent classification of Katz and Tiep when \(f(x)=x^d,\ d\in \mathbb{N}\) is a monomial. Next we report our classification when \(f(x)=(x-\alpha)^d(x-\beta)^e,\ \alpha\neq\beta\in\mathbb{F}_q,\ d,e\in\mathbb{N}\) is of Belyi type.

Sebastian Petersen (Institut für Mathematik, Universität Kassel) Finiteness properties of torsion fields of abelian varieties (12/11/2024)

(joint work with Wojciech Gajda)

Let \(A\) be an abelian variety over a field \(K\) and \(K_\mathrm{tor}\) the field obtained from \(K\) by adjoining the coordinates of all torsion points of \(A\). One may ask in which circumstances the Galois group \(\mathrm{Gal}(K_{\mathrm{tor}}/K)\) is topologically finitely generated. The answer is "yes" when \(K\) is a local field or a global function field. The answer is "no" when \(K\) is a number field. It can be shown, however, that in the number field case \(\mathrm{Gal}(K_{\mathrm{tor}}/K)\) contains an open normal subgroup whose commutator group is topologically finitely generated. Based on these observations we can answer a question of Habegger about the maximal exponent \(e\) quotient of \(\mathrm{Gal}(K_{\mathrm{tor}}/K)\) that is mentioned in a recent paper of Checcoli and Dill. There exist further applications of our finite generation results in the theory of varieties with the weak Hilbert property.

Daniel Gil Muñoz (Charles University) Module structure of the ring of integers in number fields (20/11/2024)

The normal basis theorem states that a finite and Galois extension possesses some element whose Galois conjugates form a basis for the top field as a vector space over the ground field, the so-called normal basis. This is equivalent to the top field being free as a module over the Galois group algebra with coefficients in the ground field. If we are talking about an extension of local or global fields, a more subtle question is whether the top ring of integers is free as a module over a suitable subring of the Galois group algebra. The answer is not affirmative in general, and in fact it constitutes a question of long-standing interest. This talk aims to present a survey on the known results related to this question and also on the generalization of this question to the non-Galois setting, which is carried out by means of Hopf-Galois theory.

Rachel Newton (King's College in London) Evaluating the wild Brauer group (27/11/2024)

The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety \(X\) into the set of its adelic points. The Brauer--Manin pairing cuts out a subset of the adelic points, called the Brauer--Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer--Manin set is empty then we say there's a Brauer--Manin obstruction to the existence of rational points on \(X\). Computing the Brauer--Manin pairing involves evaluating elements of the Brauer group of \(X\) at local points. If an element of the Brauer group has order coprime to \(p\), then its evaluation at a \(p\)-adic point factors via reduction of the point modulo \(p\). For \(p\)-torsion elements this is no longer the case: in order to compute the evaluation map one must know the point to a higher \(p\)-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at \(p\)-adic points gives a filtration which we describe using work of Bloch and Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer--Manin obstruction. This is joint work with Martin Bright.

Nicolas Billerey (Université Clermont Auvergne) On Darmon's program for the generalized Fermat equation of signature (r,r,p) (09/12/2024)

I will discuss a new approach toward the resolution of certain generalized Fermat equations of signature (r,r,p) which is based on the "multi-Frey" technique and on ideas from Darmon's program.

Abstracts for the Work in Progress Seminar

Tim Seuré CKKS & a novel boostrapping approach (03/12/2024)

In cryptography, homomorphic encryption (HE) enables operations like addition and multiplication to be performed directly on encrypted data. This allows complicated computations to be securely outsourced to untrusted parties, who can work with the data without ever accessing it in its decrypted form. In this seminar, I will begin by introducing the CKKS scheme, a HE scheme designed for operations on encrypted complex vectors. We will see that one limitation of this scheme is that encrypted vectors can only undergo a limited number of multiplications before requiring a refresh through the slow process known as “bootstrapping.” In the second half of this talk, I will present a novel bootstrapping method that is expected to outperform state-of-the-art algorithms for certain parameter sets.

Last modification: December 10, 2024.