Academic Year 2018/2019 - Number Theory Seminars

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

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

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


Luxembourg Number Theory Seminar

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


Work in Progress Seminar

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


Research Seminar: Topics in Computational Number Theory

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)


Collection of abstracts

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.


Last modification: 6 December 2018.