On the Uniform bound conjecture over function fields

Supervision

Carlo Gasbarri (IRMA, Strasbourg)
Steven Lu (UQAM, Montreal, Canada)

Laboratory and team

IRMA, Strasbourg - Team "AGA"

Subject description

Let K be a number field, one of the most celebrated theorems on the arithmetic of varieties over it is the Faltings Theorem (former Mordell Conjecture): One expects that the following conjecture is true: (Uniform Bound Conjecture) Let K be a number field and g≥2. There exists a number N(K,g) depending only on K and g such that, for every smooth projective curve X of genus g and defined over K, we have that Card(X(K))≤ N(K,g). In higher dimension a well established conjecture is the Lang conjecture: (Lang Conjecture) Let K be a number field, then for every smooth projective variety X defined over K and of general type, the set X(K) of the K--rational points of X is not Zariski dense. The Uniform bound conjecture is implied by the Lang Conjecture: (Caporaso, Harris, Mazur) Lang Conjecture implies the Uniform Bound Conjecture. The aim of this thesis is to study some instances of the implication in the function fields context, by a deeper analysis of the strategy used by Caporaso, Harris and Mazur which takes in account the presence of isotrivial varieties. Some facility with basics of birational geometry will certainly be very useful in this investitation.

Related mathematical skills

The successfull candidate should have a good background in basic algebraic geometry (varietes, schemes, moduli spaces, cohomology...) basic knowledge in arithmetic geometry (Lang conjecture, integral and rational points of schemes and varieties, models of varieties over a basis...).

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.