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Unlikely Intersection and Uniform Bounds for Points

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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Topic(s)

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Funding Scheme

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ERC-STG - Starting Grant

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Call for proposal

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(opens in new window) ERC-2020-STG

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Host institution

GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER
Net EU contribution

Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.

€ 1 364 063,50
Address
WELFENGARTEN 1
30167 Hannover
Germany

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Region
Niedersachsen Hannover Region Hannover
Activity type
Higher or Secondary Education Establishments
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Total cost

The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.

€ 1 364 063,50

Beneficiaries (2)

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