Project description
Uniform bounds on rational points of algebraic varieties under study
The EU-funded UnIntUniBd project aims to study the uniform bounds on rational and algebraic points. These include Mazur’s conjecture on the number of points on curves, which implies the following two strong bounds: the number of rational points on a smooth projective curve of genus g of at least 2 defined over a number field of degree d is bounded above in terms of g, d and the Mordell-Weil rank; and the number of algebraic torsion points on a smooth projective curve of genus g at least 2 is bounded above only in terms of g. The project also aims to generalise the first bound to higher-dimensional subvarieties of abelian varieties, and ultimately extend the bounds to semi-abelian varieties. As key techniques for the project, functional transcendence and unlikely intersection problems will also be investigated.
Objective
I propose to investigate the following long expected but widely open uniform bounds on rational and algebraic points. (1) Mazur’s conjecture on the number of points on curves, which implies the following two strong bounds: (1.i) the number of rational points on a smooth projective curve of genus g at least 2 defined over a number field of degree d is bounded above in terms of g, d and the Mordell- Weil rank; (1.ii) the number of algebraic torsion points on a smooth projective curve of genus g at least 2 is bounded above only in terms of g. (2) Generalize the bound in (1) to higher dimensional subvarieties of abelian varieties. (3) Extend the bounds to semi-abelian varieties. Compared with existing results, the Faltings height is no longer involved in the bounds. The proofs I propose are via Diophantine estimates. Functional transcendence and unlikely intersections on mixed Shimura varieties play important roles in the proofs. Hence as pre-requests and extensions of the three goals listed above, I will also continue investigating on functional transcendence and unlikely intersection theories as well as their potential other interesting applications in Diophantine geometry.
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Funding Scheme
ERC-STG - Starting GrantHost institution
30167 Hannover
Germany