Bordeaux, June 4–6 2025
The final conference of ANR JINVARIANT will be held in Bordeaux, at the Institut de Mathématiques de Bordeaux, on June 4–6, 2025. This will also be the opportunity to celebrate our friend and colleague Yuri Bilu and his work on the occasion of his 60th birthday.
The conference will center around: the j-invariant function and singular moduli, modular curves, integral points, Diophantine equations.
Speakers
- Yann Bugeaud
- Sara Checcoli
- Jean-Marc Deshouillers
- Jean Gillibert
- Sanoli Gun
- Philipp Habegger
- Florian Luca
- David Masser
- Joël Ouaknine
- Pierre Parent
- Amalia Pizarro Madariaga
- Marusia Rebolledo
Organizing committee
Venue
The conference will take place in the main conference hall of the IMB. The IMB is located near the Talence centre-Forum stop of tram line B; see here for information on how to access the building.
For accommodation, participants should directly book a hotel of their own choosing.
Warning!
There have been instances of scams targeted at participants in scientific conferences. Beware of anyone posing as an organizer or asking for money.
Schedule
| Wednesday 4 | Thursday 5 | Friday 6 | |
| 10:00–10:30 | Coffee | Coffee | Coffee |
| 10:30–11:30 | Masser | Bugeaud | Gun |
| 11:30–13:30 | Lunch | Lunch | Lunch |
| 13:30–14:30 | Gillibert | Ouaknine | Pizarro |
| 14:30–15:00 | Coffee | Coffee | Coffee |
| 15:00–16:00 | Rebolledo | Checcoli | Luca |
| 16:15–17:15 | Deshouillers | Habegger | Parent |
| 17:30 | Social event | Verre de l'amitié | |
| 19:30 | Conference dinner Le Café Français |
Abstracts
Search bounds for diophantine equations
David Masser (University of Basel)
Given a general polynomial equation in several variables, it is of course impossible to determine if it has a solution in integers or not. But in many interesting special cases one can do this by finding an explicit B such that there is a solution if and only if there is a solution with integers of absolute value at most B. We survey some of what is already known in this area, and we mention our recent search bound for cubic equations F(x,y) = 0 over Z; the B is very large but not quite absurdly so. On the way towards this we run into a problem about singular solutions.
Counting rational points on elliptic curves over function fields
Jean Gillibert (University of Toulouse)
Combining 2-descent techniques with Riemann-Roch and Bézout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic (and more generally hyperelliptic) curves over function fields of arbitrary characteristic. We give applications to counting S-integral points, and to bounding the 3-torsion subgroup of the Jacobian of a hyperelliptic curve over a finite field. This is joint work with Emmanuel Hallouin and Aaron Levin.
Atelier perles (beading workshop)
Marusia Rebolledo (Clermont Auvergne University)
We will see how to arrange the pearls nicely to make necklaces*. A joint work with Christian Wuthrich.
Co-primality of consecutive elements from sequences with polynomial growth
Jean-Marc Deshouillers (Institut de Mathématiques de Bordeaux)
The study of the coprimality of elements from sequences with polynomial growth (e.g. (⌊αn⌋)n or SPS sequences (⌊nc⌋)n) goes back to Evans (1953), Lambek and Moser (1955). I shall give a short outline of recent developments obtained jointly with Michael Drmota and Clemens Müllner, with Sunil L. Naik, and with Henryk Iwaniec.
Effective approximation to algebraic complex numbers by algebraic numbers of lower degree
Yann Bugeaud (University of Strasbourg)
It follows from a Liouville-type inequality that a given complex algebraic number ξ cannot be well approximated by algebraic numbers of bounded degree. When ξ is real, this result has been considerably refined by Roth in 1955, but his result is ineffective. Around 1970, Feldman introduced an important refinement in Baker's theory of lower bounds for linear forms in the logarithms of algebraic numbers, who allowed him to get an effective improvement of the Liouville-type result. We discuss these achievements and, subsequently, we focus on the case of complex non-real algebraic numbers, which has been much less studied.
Fragments of Hilbert's Program
Joël Ouaknine (Max Planck Institute for Software Systems)
Hilbert's dream of mechanising all of mathematics was dealt fatal blows by Gödel, Church, and Turing in the 1930s, almost a hundred years ago. Paradoxically, assisted and automated theorem proving have never been as popular as they are today! Motivated by algorithmic problems in discrete dynamics, non-linear arithmetic, and program verification, we examine the decidability of various logical theories over the natural numbers, and discuss a range of open questions at the intersection of logic, automata theory, and number theory.
When small points have their good reasons
Sara Checcoli (Grenoble Alpes University)
The height of an algebraic number is a real-valued function that measures the "arithmetic complexity" of the number. While numbers of height zero are well understood, many questions remain open regarding numbers of small height. For example, a key question is whether a given infinite algebraic extension of the rationals contains numbers of arbitrarily small (but non-zero) height.
In the first part of the talk, I will address the case where the answer is negative and give an overview of known results in this direction.
The main focus of the talk will then shift to the case where the answer is positive and, in particular, to the following question: in fields where small points can “obviously” be found, do these points have always "good reasons" to be small?
For instance, the field generated over the rationals by all roots of 2 contains some obvious points of very small height (0, small fractional powers of 2 multiplied by roots of unity). Does it contain other small points? A very particular case of a conjecture of Rémond suggests that the answer is no. Rémond’s conjecture more generally concerns the saturated closure of subgroups of finite ranks in tori and abelian varieties defined over number fields. It remains widely open and generalizes several important problems, such as Lehmer’s conjecture. Recently, Pottmeyer established a necessary group-theoretical condition for the conjecture to hold and proved it in the case of tori. I will present joint work with G. A. Dill, where we extend this result by showing that the condition is also satisfied for split semi-abelian varieties.
Specializing Linear Recurrence Sequences at Roots of Unity
Philipp Habegger (University of Basel)
The Skolem-Mahler-Lech Theorem characterizes the vanishing members of a linear recurrence sequence. Often only finitely many sequence members vanish and it is well understood when this is not the case. This talk concerns linear recurrence sequences of rational functions over a number field. Bilu-Luca and Ostafe-Shparlinski proved finiteness results for the sequence members that vanish when specialized at a root of unity. I will report on a new finiteness result for linear recurrence sequences of order 3. In many cases, at most finitely many sequence members vanish at a root of unity. The most difficult case involves Blaschke products, where we require linear forms in logarithms and some o-minimal geometry.
This is joint work in progress with David Masser and Alina Ostafe.
On extremal values of L-functions
Sanoli Gun (The Institute of Mathematical Sciences, Chennai)
In this talk, I will discuss about some recent results on extremal values of L-functions. This is a joint work with Rashi Lunia.
Adventures with Singular Moduli: A Survey of Joint Work with Yuri Bilu
Amalia Pizarro Madariaga (University of Valparaíso)
In this talk, we will survey several lines of research from my joint work with Yuri Bilu concerning the arithmetic properties of singular moduli. We will review our contributions to Riffaut's conjecture, including the study of singular moduli in trinomial equations, the arithmetic of rational products of singular moduli and their multiplicative structures, and the geometric distribution of CM points.
On the distance between factorials and repunits
Florian Luca (Stellenbosch University)
We show that if n ≥ n0, b ≥ 2 are integers, p ≥ 7 is prime and
Towards some equation-free quadratic Chabauty
Pierre Parent (Institut de Mathématiques de Bordeaux)
After the pioneering work of Minhyong Kim, J. Balakrishnan and others, recent years have the seen the development of radical improvements of the classical Chabauty method for determining rational points on curves, called "non-abelian Chabauty". Whereas the first formulations relied strongly on representation-theoretical tools, B. Edixhoven and G. Lido have subsequently proposed a version which is close in spirit to Chabauty's genuine geometric ideas.
All forms however demand explicit equations for the curves to be studied. In a large joint project with S. Hashimoto, K. Khuri-Makdisi, G. Lido, D. Lombardo and N. Mascot, we endeavour to formulate the geometric method in moduli terms only, in order to study new examples of modular curves, hopefully with increased efficiency–among which curves some are particularly close to Yuri's heart.
Confirmed participants
| Bill Allombert | CNRS / Institut de Mathématiques de Bordeaux |
| Cécile Armana | Laboratoire de Mathématiques de Besançon |
| Pascal Autissier | Institut de Mathématiques de Bordeaux |
| Maiken Balman Gravgaard | Aarhus University |
| Denis Benois | Institut de Mathématiques de Bordeaux |
| Attila Berczes | University of Debrecen |
| Margaret Bilu | École Polytechnique |
| Yuri Bilu | Institut de Mathématiques de Bordeaux |
| John Boxall | LMNO, Université de Caen-Normandie |
| Yann Bugeaud | Université de Strasbourg |
| Francesco Campagna | Université Clermont Auvergne |
| Sara Checcoli | Institut Fourier, Université Grenoble Alpes |
| Jean-Marc Deshouillers | Institut de Mathématiques de Bordeaux |
| Andreas Enge | INRIA / Institut de Mathématiques de Bordeaux |
| Oumar Fall | Cheikh Anta Diop University |
| Guy Fowler | University of Manchester |
| Jean Fresnel | Institut de Mathématiques de Bordeaux |
| Lorenzo Furio | IMJ-PRG |
| Stevan Gajović | Max Planck Institute for Mathematics in Bonn |
| Desirée Gijón Gómez | University of Copenhagen |
| Jean Gillibert | Institut de Mathématiques de Toulouse |
| Richard Griffon | Université Clermont Auvergne |
| Sanoli Gun | The Institute of Mathematical Sciences Chennai |
| Philipp Habegger | University of Basel |
| Lajos Hajdu | University of Debrecen |
| Suhita Hazra | Chennai Mathematical Institute |
| Jonathan Jenvrin | Institut Fourier, Université Grenoble Alpes |
| Florent Jouve | Institut de Mathématiques de Bordeaux |
| Shu Kawaguchi | Kyoto University |
| Simon Kristensen | Aarhus University |
| Samuel Le Fourn | Université Grenoble Alpes |
| Guido Maria Lido | University of Rome Tor Vergata |
| Qing Liu | Institut de Mathématiques de Bordeaux |
| Florian Luca | Stellenbosch University |
| Rashi Lunia | Max Planck Institute for Mathematics in Bonn |
| Elvira Lupoian | University College London |
| Sophie Marques | Stellenbosch University |
| David Masser | University of Basel |
| Michel Matignon | Institut de Mathématiques de Bordeaux |
| Luca Mauri | University of Pisa |
| Diana Mocanu | Max Planck Institute for Mathematics in Bonn |
| Joël Ouaknine | Max Planck Institute for Software Systems |
| Aurel Page | INRIA / Institut de Mathématiques de Bordeaux |
| Pierre Parent | Institut de Mathématiques de Bordeaux |
| Fabien Pazuki | University of Copenhagen |
| Amalia Pizarro Madariaga | University of Valparaíso |
| Purusottam Rath | Chennai Mathematical Institute |
| Marusia Rebolledo | Université Clermont Auvergne |
| Damien Robert | INRIA / Institut de Mathématiques de Bordeaux |
| Dhananjaya Sahu | Indian Institute of Technology Delhi |
| René Schoof | University of Rome Tor Vergata |
| William Stephenson | University of Manchester |
| Peter Stevenhagen | Leiden University |
| Papiya Sur | The Institute of Mathematical Sciences Chennai |
| Valerio Talamanca | Roma Tre University |
| Adrian Alexander Ticona Delgado | Institut de Mathématiques de Toulouse |
| Florian Tilliet | Université Clermont Auvergne |
| Dajano Tossici | Institut de Mathématiques de Bordeaux |
| Emanuele Tron | Institut de Mathématiques de Bordeaux |
| John Voigt | University of Sydney |
