Sinnou David was at UMI ReLaX from 01/09/2017 to 31/08/2018

My activities during this periode was centered around 6 parts:

On my own research, I proceeded to write up two results namely

(a) *On the bounded height conjecture for the torsion of abelian varieties defined over a function field*

(b) *On the Lhemer problem for semi-abelian varieties*

Project (a) is a collaboration with Prof. A. Pacheco (UFRJ, Rio de Janeiro, Brasil). It is a long standing problem dating back to the early 20th century to ensure that the torsion of an abelian variety defined on a global field should be bounded solely in terms of its field of rationality (independently of its moduli). It was solved in the mid-nineties for elliptic curves by Hindry-Silverman (function fields) and Merel (number fields), while the higher dimensional case remained widely open. We developped a novel approach to tackle the higher dimensional situation in the function field case, using diophantine gemeotry on the toroidal compactification over the integers (as developped by Chai-Faltings). The proof goes through quite well except an unnecessary restriction on the dimension of the variety compared to the residue field (due to the use of a criterion of Ogg). In order to be completely exhaustive, we would like to get rid of this restriction. We worked on it during A. Pacheco’s visit to ReLaX (Jan-Feb 2018, one month). We do expect to have a patch ready in the near future. This is a long project and the first draft (including the present restriction runs over little under 100 pages).

Project (b) also addresses a long standing problem. It is well known that the Lehmer problem generalizes naturally to any multiplicative group or any abelian variety, with essentially the same statements. Thus one would expect that the “right” statement deals with a general semi-abelian variety. Unfortunately any natural extension of the statement are plain wrong (there are obvious counterexamples) as soon as we are not in the isotrivial case anymore. This project tries to address this question. Firstly, we unify the existing counterexamples, and introduce an obstruction set (though not Zariski closed, it is a “small set”), which happens to be empty in the isotrivial case. We then state a conjecture saying that Lehmer should hold outside this obtruction set in any semi-abelian variety. We proceed to show that statement “up to an epsilon” (a Dobrowolski type of estimate) for semi-abelian varieties who have a small dimensional toric and abelian part (less than 5). The draft runs around 50 pages and we are working to have a completely satisfactory statement with no restriction on the dimension. This requires a new zero estimate. Our challenge for 2019 would be to complete this work (first state a reasonably good zero estimate, and second prove it).

2) Doctoral level courses/seminar:

– I gave two doctoral level courses, each on one semester. The first was a full proof (following Faltings, Vojta, Bombieri *et al.*) of the former Mordell conjecture (semester one). The second was an introduction to Zilber-Pink conjectures (following the work of Bombieri, Masser, Zannier, Habegger, Rémond, Viada, Maurin *et al.*). These courses required advanced knowledge of arithmetic geometry which is not readily available in textbooks. A few very talented students from the CMI part of ReLaX completed both courses.

– I ran a seminar (whih was mostly a reading group) to help the doctoral level students read a few research papers by dividing them and asking the students to lecture. This was mainly followed by young fellows from the IMSc part of ReLaX, although a handful of CMI students also participated.

3) Master’s thesis

I guided a student from CMI, S. Sahu, for his master’s thesis on the Bogomolov property for fields defined by the torsion of an elliptic curve. That thesis was submitted in June 2018.

4) Visits

We received 3 visits:

– Prof. Pacheco (scientific collaboration, two lectures at IMSc, January-February 2018).

– Prof. A. Oancea (UPMC, mini course at CMI, January 2018).

– Prof. Virginie Bonaillie (ENS, set of lectures, January 2018).

5) Internships: I guided Aritriya Mukhapadhyay’s internship (starting in July 2018). This internship, which extends into the academic year 2018-2019, consists in the study of 3 deep papers (Colliot-Thelène-Harari, Habegger, Roessler).

6) Phd programs. I also advised four students about their future doctoral projects. One, A. Kundu, was admitted in Masters at Paris-Sorbonne and I am following him. The second, S. Bakhta, wanted to study analytic number theory and was introduced to H. Helfgott (Goettingen, one of the leading experts). The third, S. Sahu, did his Masters with me, was admitted in the PhD program at Sorbonne University but finally decided to resign and join the PhD program at Bonn University. I expect the last one, A. Mukhapadhyay, to join a French graduate program next academic year.