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

- The
**Luxembourg Number Theory Seminar**hosts invited speakers. - In the
**Work in Progress Seminar**the group discusses its work in progress; it alternates with the Number Theory Seminar. - In the
**Research Seminar**the group members study a topic together; during term time the seminar takes place weekly.

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

During the Summer Semester, this seminar typically meets on __Tuesdays at 10:00__ in hybrid format in room __MNO 1.030__.

During the Winter Semester, 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.

tr>

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 | A modular construction of unramified p-extensions of Q(N^{1/p}) |

15/12/2021, 11:00-16:00 | Emiliano Ambrosi, Riccardo Brasca, Christophe Cornut, Yukako Kezuka | Luxembourg Number Theory Day |

17/01/2022, 10:30 (online) | Antigona Pajaziti | Orders of reductions of an elliptic curve in arithmetic progressions |

19/04/2022, 10:00 | Joaquín Rodrigues Jacinto | Solid locally analytic representations of p-adic Lie groups |

05/05/2022, 10:00 | Nirvana Coppola | Coleman integrals over number fields: a computational approach |

10/05/2022, 10:00 | John Bergdall | Recent investigations of L-invariants of modular forms |

17/05/2022, 10:00 | Eleni Agathocleous | On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle |

24/05/2022, 10:00 | Pip Goodman | Superelliptic cuves with large Galois images |

31/05/2022, 10:00 | Francesc Fité | On a local-global principle for quadratic twists of abelian varieties |

13/06/2022, 15:00 (online) | Félix Baril Boudreau | Reduction of L-functions of Elliptic Curves Modulo Integers |

29/06/2022, 14:00 (MNO 1.040) | Shaunak Deo | The Eisenstein ideal of weight k and ranks of Hecke algebras |

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) |

The seminar will take place on __Thursdays from 11:00 to 12:30__ in hybrid format. Here is the program of the seminar.

Date (Room) | Speaker | Title |

17/03/2022, 11:00 (MSA 4.060) | Alexandre Maksoud | Overview |

24/03/2022, 11:00 (MSA 4.060) | Alfio Fabio La Rosa | Relative Lubin-Tate group laws |

31/03/2022, 11:00 (MSA 4.060) | Alfio Fabio La Rosa | Lubin-Tate extensions and local class field theory |

07/04/2022, 11:00 (MSA 4.060) | Flavio Perissinotto | Witt vectors |

03/05/2022, 9:45 (MNO 1.030) | Bryan Advocaat | (phi,Gamma)-modules |

05/05/2022, 11:00 (MSA 4.060) | Emiliano Torti | Overconvergent (phi,Gamma)-modules |

19/05/2022, 11:00 (MSA 4.180) | Bryan Advocaat | Fontaine's rings |

02/06/2022, 11:00 (MSA 4.060) | Bryan Advocaat | Fontaine's rings, II |

02/06/2022, 11:30 (MSA 4.060) | Andrea Conti | Crystalline and semistable representations |

??/??/2022, 11:00 (MSA 4.060) | tba | Cohomology of (phi,Gamma)-modules |

??/??/2022, 11:00 (MSA 4.060) | tba | Iwasawa cohomology |

The seminar will take place on __Thursdays 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 |

28/02/2021, 14:00 (MNO 1.010) | Mattia Cavicchi (IRMA Strasbourg) | Intersection Cohomology and Shimura Varieties (closing talk; abstract below) |

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

- 1. The zeta function is rational.

- 2. It satisfies a functional equation.

- 3. It satisfies an analogue of the Riemann Hypothesis.

- 4. The degrees of the polynomials appearing in the factorisation of the zeta function are the Betti numbers of an appropriate analytification of our variety.

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.

Meeting number: 2733 415 1759 Pwd: MspGzFi3r93

**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.

**Jaclyn Lang (Temple University)** *A modular construction of unramified p-extensions of Q(N^{1/p})*

In his 1976 proof of the converse of Herbrand's theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree p-extensions of the p-th cyclotomic field when p is an odd prime. After reviewing Ribet's strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree p-extensions of Q(N^{1/p}) when N is a prime that is congruent to -1 mod p. This answers a question posed on Frank Calegari's blog.

**Antigona Pajaziti (Sabanci University Istanbul)** *Orders of reductions of an elliptic curve in arithmetic progressions*

Let E be an elliptic curve defined over a number field K with ring of integers R. We consider the set S of all the orders of reductions of E modulo the primes of R. Given an integer m > 1, one may ask how many residue classes modulo m have an intersection of positive density with S. Using results of Serre and Katz, we show that there are at least two such residue classes; except for explicit families of elliptic curves and corresponding values of m. We then describe this exceptional set of elliptic curves and list the values of m when K is of degree at most 3 or K is Galois of degree 4.

**Mattia Cavicchi (IRMA Strasbourg)** *Intersection Cohomology and Shimura Varieties*

Intersection cohomology was defined by Goresky and MacPherson as a substitute of singular cohomology, coinciding with it on topological manifolds, and still satisfying Poincaré duality for (a suitable class of) singular topological spaces. It was then recovered in the more general and abstract framework of the theory of perverse sheaves, as introduced by Beilinson, Bernstein, Deligne and Gabber, and has since proven to be an invariant of fundamental importance at the crossroads of topology, algebraic geometry and representation theory. The aim of this talk is to recall this circle of ideas and to give an example of their role in arithmetic geometry, through the theory of weights and the study of the intersection cohomology of Shimura varieties.

**Joaquin Rodrigues Jacinto (Université Paris-Saclay)** *Solid locally analytic representations of p-adic Lie groups*

I will explain how to reformulate the theory of locally analytic representations of a compact p-adic Lie group developed by Schneider and Teitelbaum in the language of condensed mathematics of Clausen and Scholze. This will allow us to prove comparison results between different types of cohomologies of these representations, extending and generalising classical results of Lazard. This is joint work with Juan Esteban Rodríguez Camargo.

**Nirvana Coppola (VU Amsterdam)** *Coleman integrals over number fields: a computational approach*

One of the deepest mathematical results is Faltings's Theorem on the finiteness of rational points on an algebraic curve of genus g >= 2. A much more difficult question, still not completely answered, is whether given a curve of genus g >= 2, we can find all its rational points, or, more in general, all points defined over a certain number field. An entire (currently very active!) area of research is devoted to find an answer to such questions, using the "method of Chabauty". In this seminar, I will talk about one of the first tools employed in Chabauty method, namely Coleman integrals, which Coleman used to compute an explicit bound on the number of rational points on a curve. After explaining how this is defined, I will give a generalisation of this definition for curves defined over number fields, and explain how to explicitly compute these integrals. This is based on an ongoing project, which started during the Arizona Winter School 2020, joint with E. Kaya, T. Keller, N. Müller, S. Muselli.

**John Bergdall (MPIM Bonn/Bryn Mawr College)** *Recent investigations of L-invariants of modular forms*

This talks focuses on recent numerical investigations of L-invariants. Mazur, Tate, and Teitelbaum discovered L-invariants in the 1980's. They sought a p-adic analogue to Birch and Swinnerton-Dyer's conjecture on elliptic curves. In the decades since, L-invariants have arisen in many further arithmetic contexts. This includes: L-functions, families of modular forms, and Galois representations. Our talk highlights computations of L-invariants carried out in a range of contexts. We further raise distributional questions. These complement open questions about the non-Archimedean behavior of Hecke eigenvalues. Our contributions are part of an ongoing joint project with Robert Pollack.

**Eleni Agathocleous (CISPA Saarbrücken)** *On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle*

Abstract here.

**Pip Goodman (MPIM Bonn)** * Superelliptic cuves with large Galois images *

Let K be a number field. The inverse Galois problem for K asks if for every finite group G there exists a Galois extension L/K whose Galois group is isomorphic to G. Many people have used torsion points on abelian varieties to realise symplectic similitude groups (GSp_n(F_\ell)) over \Q. In this talk, we examine mod \ell Galois representations attached to superelliptic curves and use them to realise general linear and unitary similitude groups over cyclotomic fields. A variety of mathematics is involved, including group theory, CM theory and models of curves.

**Francesc Fité (Universitat de Barcelona)** *On a local-global principle for quadratic twists of abelian varieties*

Let A and A' be abelian varieties defined over a number field k. In the talk I will consider the following question: Is it true that A and A' are quadratic twists of one another if and only if they are quadratic twists modulo p for almost every prime p of k? Serre and Ramakrishnan have given a positive answer in the case of elliptic curves and a result of Rajan implies the validity of the principle when A and A' have trivial geometric endomorphism ring. For not necessarily simple abelian varieties, I will show that the answer is affirmative up to dimension 3, but that it becomes negative in dimension 4. The proof builds on Rajan's result and uses a Tate module tensor decomposition of an abelian variety geometrically isotypic (the latter obtained in collaboration with Xavier Guitart).

**Félix Baril Boudreau (Western University, Canada)** *Reduction of L-functions of Elliptic Curves Modulo Integers*

Let F_q be a finite field of size q where q is a power of a prime p \geq 5. Let C be a smooth, proper, and geometrically connected curve over F_q. Consider an elliptic curve E over the function field K of C with nonconstant j-invariant. One can attach to E its L-function L(T,E/K), which is a generating function that contains information about the reduction types of E at the different places of K. The L-function of E/K was proven to be a polynomial in Z[T].

In 1985, Schoof devised an algorithm to compute the zeta function of an elliptic curve over a finite field by directly computing its numerator modulo sufficiently many primes \ell. By analogy with Schoof, we consider an elliptic curve E over K with nonconstant j-invariant and study the problem of directly computing the reduction of L(T,E/K) modulo \ell. In this work, we obtain results in two directions. Firstly, given an integer N different from p and an elliptic curve E with K-rational N-torsion, we give a formula for the reduction modulo N of the L-function of certain quadratic twists, extending a result of Chris Hall. We also give a formula relating the L-functions modulo 2 of any two quadratic twists of E, without any assumptions on the K-rational 2-torsion. Secondly, given a prime \ell \neq p, we give, under some relatively general conditions, formulas for the reduction of L(T,E/K) modulo \ell. The formulas in this work are amenable to computation by algorithms that are more efficient than naive point-counting methods.

**Shaunak Deo** *The Eisenstein ideal of weight k and ranks of Hecke algebras*

Let $p$ and $l$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. Using deformation theory of Galois representations, we will give a necessary and sufficient condition for the $Z_p$-rank of the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $\Gamma_0(\ell)$ at the maximal Eisenstein ideal containing $p$ to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes.

Last modification: 29 June 2022.