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 (Winter Semester 2024)

Luxembourg Number Theory Seminar (Summer Semester 2025)

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.

Chantal David (Université de Concordia) Rank and non-vanishing in the family of elliptic curves \(y^2 = x^3 - dx\) (25/02/2025)

The elliptic curves \(E_d : y^2 = x^3 - dx\), where \(d\) is a 4th power-free integer, form a family of quartic twists. We study in this talk the average analytic rank \(r(d)\) over the family. Under the GRH, we show that the average analytic rank is bounded by \(13/6\), and by \(3/2\) assuming a conjecture of Heath-Brown and Patterson about the distribution of quartic Gauss sums. Since the same result holds when we restrict to the subfamilies of curves \(E_d\) where the root number is fixed (i.e. \(W(E_d) = \pm 1\)), this shows that there is a positive proportion of curves with \(r(E_d) = 0\) among the curves with even analytic rank, and a positive proportion of curves with \(r(E_d) = 1\) among the curves with odd analytic rank.
Our results are similar to the results obtained by Heath-Brown for the analytic rank of the quadratic twists \(dy^2 = x^3 + ax + b\) under the GRH. For the quadratic twists, it was shown in the recent ground-breaking work of Smith that half of the quadratic twists have algebraic rank \(0\) and half of the quadratic twists have algebraic rank \(1\), under the assumption that the Tate-Shafarevic group is finite. For the case of the quartic twists \(E_d : y^2 = x^3 - dx\), no bound for the average algebraic rank is known.
This is joint work with L. Devin, A. Fazzari and E. Waxman.

Paul Péringuey (University of British Columbia) Refinements of Artin's primitive root conjecture (25/03/2025)

Let \(\rm{ord}_p(a)\) be the order of \(a\) in \( \left(\mathbb{Z}/p\mathbb{Z} \right)^*\). In 1927, Artin conjectured that the set of primes \(p\) for which an integer \(a\neq -1,\square\) is a primitive root (i.e. \(\rm{ord}_p(a)=p-1\)) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH).

In this talk we will study the behaviour of \(\rm{ord}_p(a)\) as \(p\) varies over primes, in particular we will show, under GRH, that the set of primes \(p\) for which \(\rm{ord}_p(a)\) is "\(k\) prime factors away" from \(p-1\) has a positive asymptotic density among all primes except for particular values of \(a\) and \(k\). We will interpret being "\(k\) prime factors away" in three different ways, namely \(k=\omega(\frac{p-1}{\rm{ord}_p(a)})\), \(k=\Omega(\frac{p-1} {\rm{ord}_p(a)})\) and \(k=\omega(p-1)-\omega(\rm{ord}_p(a))\), and present conditional results analogous to Hooley's in all three cases and for all integer \(k\). From this, we will derive conditionally the expectation for these quantities.

Furthermore we will provide partial unconditional answers to some of these questions.

This is joint work with Leo Goldmakher and Greg Martin.

Douglas Ulmer (University of Arizona) p-torsion of curves in characteristic p (01/04/2025)

The Torelli locus---the image of the moduli space of curves (Mg) in the moduli space of abelian varieties (Ag)---is much-studied but still mysterious. In characteristic p, Ag has a beautiful stratification by the isomorphism type of A[p], and examples show that Mg is far from transverse to this stratification. In an ongoing project, we develop tools to understand (and perhaps make principled conjectures about) which strata of Ag meet Mg. In this talk, we explain some of the structures involved (including a crash course on finite group schemes) and then give some new results about them. Parts of this are joint work with Bryden Cais and Rachel Pries.

Emmanuel Kowalski (ETH Zürich) Equidistribution for polynomial periods and Wasserstein distances (22/04/2025)

(Joint work with T. Untrau)

There are many equidistribution theorems in modern number theory, which provide statistical regularity properties of otherwise seemingly random arithmetic objects. A topic of particular interest is to quantify some of these theorems, in terms of speed of convergence in a suitable sense. We will present one special case related to analogues for general polynomials of the distribution of cyclotomic Gaussian periods, and explain how the Wasserstein distances, familiar to analysts and probabilists, provide a particularly convenient framework to quantify this result.

Juan Carlos Hernández Bocanegra (CINVESTAV) Extended genus fields of abelian extensions of rational function fields (29/04/2025)

We obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a composite of two cyclotomic extensions of a global rational function field is equal to the composite of their respective extended genus fields, to obtain our main result. This result is that the extended genus field of a general finite abelian extension of a global rational function field, is given explicitly in terms of the field and of the extended genus field of its "cyclotomic projection".

Himanshu Shukla (University of Bayreuth) On the factorization of twisted L-values and \(11\)-descents over \(C_5\) extensions. (06/05/2025)

Let \(K/\mathbb{Q}\) be a cyclic extension of \(\mathbb{Q}\) and \(E/\mathbb{Q}\) be an elliptic curve. In this joint work with Celine Maistret (University of Bristol) we investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character \(\chi\) factoring through \(K\) we give an explicit conjecture relating the ideal factorization of \(L(E,\chi,1)\) to the Galois module structure of \(\Sha(E/K)\). We provide numerical evidence for this conjecture using the methods of visualization and a result of Greenberg and Vatsal on \(p\)-adic \(L\)-functions, and \(p\)-descent. For the latter, we present a procedure that makes performing an \(11\)-descent over a \(C_5\) number field practical for an elliptic curve \(E/\mathbb{Q}\) with complex multiplication.

Antonio Rojas León (Universidad de Sevilla) Equidistribution and independence of Gauss sums (13/05/2025)

Given a finite field \(k/\mathbb{F}_p\) and \(n\) \(r\)-tuples \(\mathbf{a}_1,\ldots,\mathbf{a}_n\in\mathbb{Z}^r\) we consider the question of the simultaneous distribution of the normalized multinomial Gauss sums \(G(\chi^{\mathbf{a}_1}),\ldots,G(\chi^{\mathbf{a}_n})\) in \((S^1)^n\) as \(\chi\) ranges on the set of \(r\)-tuples of characters of \(k^\times\). This is a wide generalization of many previous equidistribution results about Gauss and Jacobi sums.

As a consequence, we show that all non-trivial relations among these sums can be expressed as combinations of the three well-known relations: Frobenius invariance, conjugacy and the Hasse-Davenport product formula.

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.

Alexandre Benoist The distribution of Elkies primes (11/03/2025)

In many applications of elliptic curves in public-key cryptography, it is necessary to determine efficiently the number of rational points of an elliptic curve defined over a finite field. For fields of cryptographic size, the best method up to date for large characteristic is the Schoof-Elkies-Atkin (SEA) algorithm. Its time complexity essentially depends on the distribution of Atkin and Elkies primes. Given an elliptic curve \(E\) defined over a finite field \(\mathbb{F}_q\), a prime \(\ell\) is said to be Elkies if there is an isogeny from \(E\) of degree \(\ell\) defined over \(\mathbb{F}_q\), otherwise it is said to be Atkin. The heuristic argument to determine the complexity of the algorithm is that there is roughly the same number of Elkies and Atkin primes. Theoretical results in this direction have been established in the last decade by Shparlinski and Sutherland in an average setting, leading to a complexity result of the SEA algorithm in average.

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). In our work, we performed numerical experiments, suggesting that the number of Elkies primes for reductions of \(E\) modulo primes converges weakly to a Gaussian distribution. We managed to prove this result and to generalize it to the setting of abelian varieties (generalization of elliptic curves in higher dimension) with real multiplication.

The first aim of the talk will be to explain how the SEA algorithm works and how its complexity depends on Elkies primes. Then, I will talk about the proof of our convergence result. Finally, I will take time to introduce my ideas to prove a similar result in another setting.

This is joint work with Jean Kieffer.

Last modification: May 12, 2025.