Summary
"The theory of rational points on curves and their Jacobians is distinguished by being both attractive and notoriously difficult. Despite major theoretical advances, explicit methods are of particular importance in this area. For instance, the conjecture of Birch and Swinnerton-Dyer (BSD), one of the Millennium Prize problems, was formulated based on numerical evidence. A proof of the strong version of this conjecture for abelian varieties seems out of reach at present, and even the verification in examples was, until recently, only possible in dimension 1.
Besides being interesting in its own right, the importance of explicit methods for the computation of the rational points on curves stems from the fact that many moduli problems can be reduced to such computations. Therefore, explicit methods can be used to solve theoretical problems, but in the other direction, theoretical advances often lead to improved explicit methods. One example is the recent computation of the rational points on the ""cursed curve"" X_ns^+(13) using the quadratic Chabauty (QC) method, an explicit special case of Kim's non-abelian Chabauty program.
We propose two research projects, connected by height theory, to significantly advance the state of the art in explicit methods for rational points on curves and Jacobians. In the first one, we will develop an explicit theory of heights to compute Mordell-Weil groups of Jacobians of non-hyperelliptic curves of genus 3. We will use it for the verification of the strong BSD conjecture for modular examples, going beyond the hyperelliptic case for the first time. In the second one, we will drastically increase the applicability of the QC method by removing several restrictive conditions, and apply it to Atkin-Lehner quotients of modular and Shimura curves, thereby solving several open moduli problems."
Besides being interesting in its own right, the importance of explicit methods for the computation of the rational points on curves stems from the fact that many moduli problems can be reduced to such computations. Therefore, explicit methods can be used to solve theoretical problems, but in the other direction, theoretical advances often lead to improved explicit methods. One example is the recent computation of the rational points on the ""cursed curve"" X_ns^+(13) using the quadratic Chabauty (QC) method, an explicit special case of Kim's non-abelian Chabauty program.
We propose two research projects, connected by height theory, to significantly advance the state of the art in explicit methods for rational points on curves and Jacobians. In the first one, we will develop an explicit theory of heights to compute Mordell-Weil groups of Jacobians of non-hyperelliptic curves of genus 3. We will use it for the verification of the strong BSD conjecture for modular examples, going beyond the hyperelliptic case for the first time. In the second one, we will drastically increase the applicability of the QC method by removing several restrictive conditions, and apply it to Atkin-Lehner quotients of modular and Shimura curves, thereby solving several open moduli problems."
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| Web resources: | https://cordis.europa.eu/project/id/101064790 |
| Start date: | 01-06-2023 |
| End date: | 31-05-2025 |
| Total budget - Public funding: | - 187 624,00 Euro |
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Original description
"The theory of rational points on curves and their Jacobians is distinguished by being both attractive and notoriously difficult. Despite major theoretical advances, explicit methods are of particular importance in this area. For instance, the conjecture of Birch and Swinnerton-Dyer (BSD), one of the Millennium Prize problems, was formulated based on numerical evidence. A proof of the strong version of this conjecture for abelian varieties seems out of reach at present, and even the verification in examples was, until recently, only possible in dimension 1.Besides being interesting in its own right, the importance of explicit methods for the computation of the rational points on curves stems from the fact that many moduli problems can be reduced to such computations. Therefore, explicit methods can be used to solve theoretical problems, but in the other direction, theoretical advances often lead to improved explicit methods. One example is the recent computation of the rational points on the ""cursed curve"" X_ns^+(13) using the quadratic Chabauty (QC) method, an explicit special case of Kim's non-abelian Chabauty program.
We propose two research projects, connected by height theory, to significantly advance the state of the art in explicit methods for rational points on curves and Jacobians. In the first one, we will develop an explicit theory of heights to compute Mordell-Weil groups of Jacobians of non-hyperelliptic curves of genus 3. We will use it for the verification of the strong BSD conjecture for modular examples, going beyond the hyperelliptic case for the first time. In the second one, we will drastically increase the applicability of the QC method by removing several restrictive conditions, and apply it to Atkin-Lehner quotients of modular and Shimura curves, thereby solving several open moduli problems."
Status
SIGNEDCall topic
HORIZON-MSCA-2021-PF-01-01Update Date
09-02-2023
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