Summary
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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| Web resources: | https://cordis.europa.eu/project/id/945714 |
| Start date: | 01-09-2020 |
| End date: | 31-08-2025 |
| Total budget - Public funding: | 1 499 916,00 Euro - 1 499 916,00 Euro |
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Original description
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.Status
SIGNEDCall topic
ERC-2020-STGUpdate Date
27-04-2024
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