The Algebra and Number Theory group of the University of Luxembourg hosts three seminars.

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

Unless announced otherwise, the seminars take place in the "work place" MNO 6B in the 6th floor of the Maison du Nombre in Esch-Belval.

Everyone is invited to attend! For more information, please contact Gabor Wiese.

Date | Speaker | Title |

30/10/2018, 14:00h | Mladen Dimitrov (Lille) | Uniform boundedness for the rational torsion of abelian 3-folds of Picard type |

30/10/2018, 15:15h | Eduardo Soto (Barcelona) | Raising the level at your favorite prime |

15/11/2018, 09:30h | Hoan-Phung Bui (Université libre de Bruxelles) | From Galois to Hopf-Galois |

20/11/2018, 11:00h | Ian Kiming (Copenhagen) | Newton polygons for the U_p operator and p-adic families of eigenforms: New explicit computations for p=5,7,13, tame level 1 |

27/11/2018, 11:00h | Jaclyn Lang (Paris) | Images of two-dimensional Galois representations |

04/12/2018, 11:00h | Paloma Bengoechea Duro (ETH Zürich) | Periods of modular functions and diophantine approximation |

09/01/2019, 15:00h | Filip Najman (Zagreb) | Growth of torsion groups of elliptic curves upon base change |

15/03/2019, 14:00h | Bas Edixhoven (Leiden) | Geometric interpretation of quadratic Chabauty |

03/05/2019, 11:30h | Shaunak V. Deo (Tata Institute, Mumbai) | Hilbert modular eigenvariety at exotic and CM classical points of parallel weight one |

04/06/2019, 14:00h | Leo Murata (Meiji Gakuin University, Japan) | On a distribution property of the multiplicative order of a(mod p) and a(mod pq) |

17/06/2019, 14:00h | Anna Somoza Henares (Leiden, Bonn) | Inverse Jacobian problem for cyclic plane quintic curves |

21/06/2019, 10:00h | Rodolphe Richard (IHES, Paris) | Towards an "arithmetic" André-Oort conjecture |

Date | Speaker | Title |

25/09/2018, 09:00 | Samuele Anni | On conjectures of Maeda and Coleman (part 1) |

02/10/2018, 10:15 | Samuele Anni | On conjectures of Maeda and Coleman (part 2) |

09/10/2018, 11:00 | Samuele Anni | On conjectures of Maeda and Coleman (part 3) |

16/10/2018, 11:00 | Emiliano Torti | On level raising modulo prime powers |

23/10/2018, 11:00 | Pietro Sgobba | Kummer Theory for Number Fields |

29/11/2018, 10:00 | Jim Barthel | On the (non-)equivalence of integral binary quadratic forms and their negative forms |

26/02/2019, 14:00 | Antonella Perucca | Kummer extensions of the rationals |

03/04/2019, 11:00 | Sebastiano Tronto | Computing degrees of Kummer extensions |

26/04/2019, 11:00 | Tara Trauthwein | Approximation of Cantor rational numbers via primitive words |

03/05/2019, 10:00 | Emiliano Torti | Local constancy modulo prime powers of reductions of 2-dimensional crystalline representations |

All references below are for Neukirch: Class Field Theory/Klassenkörpertheorie, Springer.

Date | Speaker | Title |

26/02/2019, 09:00 | Fritz Hörmann | Overview |

26/02/2019, 11:00 | Alexander Rahm | I.1-I.3 |

05/03/2019, 11:00 | Alexander Rahm | I.4-I.5 |

12/03/2019, 09:00 | Luca Notarnicola | I.6+I.7 |

12/03/2019, 14:00 | Emiliano Torti | II.1 |

26/03/2019, 11:00 | Sebastiano Tronto | II.2+II.3 |

02/04/2019, 09:00 | Alisa Govzmann | II.4 |

02/04/2019, 11:00 | Pietro Sgobba | II.5+II.6 |

23/04/2019, 10:30 | Daniel Berhanu Mamo | II.7 (120 minutes) |

30/04/2019, 09:00 | Alisa Govzmann | III.1+III.2 |

30/04/2019, 11:00 | Sebastiano Tronto | III.3 |

07/05/2019, 09:00 | Luca Notarnicola | III.4 |

07/05/2019, 11:00 | Pietro Sgobba | III.5 |

14/05/2019, 11:00 | Gabor Wiese | III.6 |

Date | Speaker | Title |

25/09/2018, 14:00 | Gabor Wiese | Congruences of modular forms and their experimental testing (part 1) |

02/10/2018, 14:00 | Gabor Wiese | Congruences of modular forms and their experimental testing (part 2) |

09/10/2018, 14:00 | Luca Notarnicola | Edwards Curves and Diffie-Hellman exchange |

16/10/2018, 14:00 | Jim Barthel | Elliptic Curves Factorisation Method |

06/11/2018, 14:00 | Sebastiano Tronto | Local-global principle for torsion |

13/11/2018, 14:00 | Emiliano Torti | Schoof's algorithm |

27/11/2018, 14:00 | Daniel Berhanu Mamo | Models of modular curves |

04/12/2018, 14:00 | Pietro Sgobba | Class field theory |

11/12/2018, 11:00 | Emiliano Torti | Serre invariants for elliptic curves (part 1) |

11/12/2018, 14:00 | Samuele Anni | Serre invariants for elliptic curves (part 2) |

**Mladen Dimitrov (University of Lille) ** *Uniform boundedness for the rational torsion of abelian 3-folds of Picard type *

In 1969 Manin proved a uniform version of Serre's celebrated result on the openness of the Galois image in the automorphisms of the p-adic Tate module of any non-CM elliptic curve over a given number field. Recently in a series of papers Cadoret and Tamagawa established a definitive result regarding the uniform boundedness of the p-primary torsion for 1-dimensional abelian families. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension, in the case of abelian families parametrized by Picard modular surfaces over an imaginary quadratic field M. Namely, we establish a uniform bound for the p-primary torsion of principally polarized abelian 3-folds with multiplication by M, but without CM factors, subject to some rationality condition at the primes dividing the discriminant of M.

**Eduardo Soto (Barcelona) ** *Raising the level at your favorite prime *

Considering chains of congruences between modular forms of various levels is a central method when proving Serre's conjecture, Taniyama Shimura, etc. In this talk I'll report a trick deduced from Ribet's level raising theorem to raise the level at any prime, e.g. your favorite prime. This is joint work with L. Dieulefait.

**Ian Kiming (Copenhagen)** * Newton polygons for the U_p operator and p-adic families of eigenforms: New explicit computations for p=5,7,13, tame level 1 *

Let p be a prime number. In 1997 R. F. Coleman proved a fantastic theorem, namely that a classical eigenform (some weight, some level $N$, say prime to $p$) of finite so-called p-slope can be embedded into a p-adic analytic family of eigenforms of the same p-slope, same level, but varying weights. In concrete terms, this implies that the given form is congruent to other eigenforms modulo higher and higher powers of p. If one wants to quantify this concrete implication (for a given form), one needs to say something explicit about the ``radius'' of the p-adic analytic family. There are precious few explicit examples of this, - before our work this was all for primes 2 and 3 and tame level 1 (Emerton, Coleman-Stevens-Teitelbaum, Smithline, ...). We provide new explicit examples for p=5,7 and tame level 1. The main thing that one has to do is to obtain explicit lower bounds for the Newton polygon of the U_p operator acting on overconvergent p-adic modular functions. We prove new bounds for the above cases, and also for p=13. The method comes from Coleman-Stevens-Teitelbaum, but there are new challenges to consider that are not so visible when p=3. This is joint work with Dino Destefano.

**Jaclyn Lang (Paris)** * Images of two-dimensional Galois representations *

There is a general philosophy that the image of a Galois representation should be as large as possible, subject to the symmetries of the geometric object from which it arose. This can be seen in Serre's open image theorem for non-CM elliptic curves, Ribet and Momose's work on Galois representations attached to modular forms, and recent work of the speaker and Conti, Iovita, Tilouine on Galois representations attached to Hida and Coleman families of modular forms. Recently, Bellaiche developed a way to measure the image of an arbitrary Galois representation taking values in GL(2) of a local ring A. Under the assumptions that A is a domain and the residual representation is not too degenerate, we explain how the symmetries of such a representation are reflected in its image. This is joint work with Andrea Conti and Anna Medvedovsky.

**Paloma Bengoechea Duro (ETH Zürich)** *Periods of modular functions and diophantine approximation *

For a real quadratic irrationality w and the classical Klein's modular invariant j, the "value" j(w) has been recently defined using the period of j along the closed geodesic associated to w in the hyperbolic plane. Works of Duke, Imamoglu, Toth, and Masri establish analogies between these values and singular moduli when they are both gathered in traces. However, the arithmetic/algebraic properties of the individual values j(w) remain inaccessible. In this talk, we will address a conjecture of Kaneko on bounds for these values. Our strategy consists in studying the values j(w) according to the diophantine properties of w. This is joint work with O. Imamoglu.

**Filip Najman (Zagreb)** *Growth of torsion groups of elliptic curves upon base change*

We study how the torsion of elliptic curves over number fields grows upon base change, and in particular prove various necessary conditions for torsion growth. In particular, we prove that E(K)_{tors}=E(Q)_{tors} for all elliptic curves E defined over Q and all number fields K of degree d, where d is not divisible by a prime greater or equal to 7. This is joint work with Enrique Gonzalez-Jimenez.

**Bas Edixhoven (Leiden)** *Geometric interpretation of quadratic Chabauty *

Joint work with Guido Lido. Chabauty's method to find all rational points on a curve C over Q of genus g > 1 is to intersect, for a suitable prime p, inside the p-adic Lie group J(Q_p) (with J the jacobian of C), the 1-dimensional p-adic manifold C(Q_p) with the closure of J(Q). This closure is a p-adic Lie group of dimension at most r, the rank of J(Q). If r < g then this works well. Minhyong Kim has a program called `nonabelian Chabauty', where deeper quotients of the fundamental group of C are exploited (J corresponds to the abelianisation). The recently developed `quadratic Chabauty method' (Balakrishnan, Dogra, Muller, Tuitman, Vonk) can treat cases where r is larger and J has sufficiently many symmetric endomorphisms, notably the `cursed curve'. In this lecture I will give a geometric description of the quadratic Chabauty method in terms of the Poincare torsor on J times its dual.

**Leo Murata (Meiji Gakuin University, Japan) ** *On a distribution property of the multiplicative order of a(mod p) and a(mod pq)*

Let a be a positive integer, p be a prime number such that (a,p)=1. We are mainly interested in the distribution properties of D_a(p), the residual order of the class a(mod p) in (Z/pZ)^*, where we fix a and p varies. Under the generalized Riemann Hypothesis(GRH), we can determine the natural density of the primes p satisfying ?D_a(p) is congruent to j (mod k)?, for any preselected congruent class j(mod k). We consider the same question for the group (Z/pqZ)^*, where p and q are distinct primes. This multiplicative group is not cyclic, but we can prove the similar result for some simple cases.

**Anna Somoza Henares (Leiden, Bonn)** *Inverse Jacobian problem for cyclic plane quintic curves*

We consider the problem of computing an equation for a curve, given only its periods (or equivalently, its analytic Jacobian). In the genus one case, this can be done by using classical Weierstrass functions and it is the key step if one wants to write down equations for elliptic curves with complex multiplication (CM). Also in higher-genus, the theory of CM tells us all potential period matrices of curves whose Jacobian has CM, and computing curve equations is the hardest part. Beyond the classical case of elliptic curves (g=1), efficient solutions of this problem are known for both genus 2 and genus 3. In this talk I will give an method that deals with the case y^5 = f(x) with deg f = 4 or 5, inspired by some of the ideas present in a previous algorithm for the family of Picard curves y^3 = f(x) with deg f = 4.

**Rodolphe Richard (IHES, Paris)** *Towards an "arithmetic" André-Oort conjecture*

We present a not trivially false generalisation of the André-Oort conjecture. Indeed we prove it in two non trivial cases (one, under GRH, j.w. Edixhoven). We relate it to, and motivate it by, recents trends in equidistribution.

Last modification: 3 July 2019.