Arithmetic of hyperelliptic curves
Lead Research Organisation:
King's College London
Department Name: Mathematics
Abstract
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Organisations
People |
ORCID iD |
| Vladimir Dokchitser (Principal Investigator) |
Publications
Betts L
(2022)
Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction
in Journal of Number Theory
BETTS L
(2018)
Variation of Tamagawa numbers of semistable abelian varieties in field extensions
in Mathematical Proceedings of the Cambridge Philosophical Society
Dickson M
(2018)
Products of Eisenstein series and Fourier expansions of modular forms at cusps
in Journal of Number Theory
DICKSON M
(2020)
Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level
in Journal of the Mathematical Society of Japan
Dokchitser T
(2022)
Arithmetic of hyperelliptic curves over local fields
in Mathematische Annalen
Dokchitser T
(2021)
Tate module and bad reduction
in Proceedings of the American Mathematical Society
Dokchitser T
(2019)
Algebraic Curves and Their Applications
Dokchitser T
(2018)
Quotients of hyperelliptic curves and Étale cohomology
in The Quarterly Journal of Mathematics
Morgan A
(2019)
Quadratic twists of abelian varieties and disparity in Selmer ranks
in Algebra & Number Theory
| Description | This was a joint project with EP/M016838/1 (University of Bristol), and some of the results described below were joint work between the two teams. The report will not address the findings that were primarily obtained by the Bristol team. The aim of the project was to modernise our approach to the arithmetic of hyperelliptic curves, by bringing in the number theoretic machinery related to L-functions and Selmer groups into the subject. The most significant achievements were the introduction of a new approach to hyperelliptic curves over local fields, and a new understanding of some of the key properties of hyperelliptic curves over number fields. The analogues of these questions in the classical setting of elliptic curves have been fundamental in the development of modern number theory. The work on hyperelliptic curves over number fields focused on the so-called parity conjecture (a weaker form of the famous Birch-Swinnerton-Dyer conjecture, which is one of the central problems in modern mathematics) and on the mysterious Tate-Shafarevich group, which in this contexts exhibits a phenomenon that does not appear in the classical case of elliptic curves. Our contributions include the study of Tate-Shafarevich groups in families of quadratic twists, the 2-parity conjecture for Jacobians of hyperelliptic curves over quadratic extensions of the base field, new cases of the parity conjecture of general abelian varieties, and a proof of the parity conjecture for semistable abelian surfaces: - Quadratic twists of abelian varieties and disparity in Selmer ranks (Algebra and Number Theory); - 2-Selmer parity for hyperelliptic curves in quadratic extensions (arxiv:1504.01960); - Variation of Tamagawa numbers of semistable abelian varieties in field extensions (Math. Proc. Cambridge Philos. Soc.); - Parity conjecture for abelian surfaces (arxiv:1911.04626). Our key contribution to the local theory is the introduction of a new mathematical tool, "cluster pictures", for hyperelliptic curves given by equations y^2=f(x). Cluster pictures only encode elementary information about the roots of f(x), but turn out to be powerful enough to let one easily recover a vast amount of arithmetic information about the curve. The theory is very much user-oriented: it lets a number theorist bypass intricate arithmetic geometry and access otherwise subtle information about a hyperelliptic curve. This was built up across several papers: - Tate module and bad reduction (arxiv:1809.10208); - Arithmetic of hyperelliptic curves over local fields (arxiv:1808.02936); - Semistable types of hyperelliptic curves (Contemporary Mathematics); - Quotients of hyperelliptic curves and etale cohomology (Quarterly Journal of Mathematics); - On the computation of Tamagawa numbers and Neron component groups of Jacobians of semistable hyperelliptic curves (arxiv.org:1808.05479). Other achievements of the project include contributions to the theory of modular and Siegel modular forms: - Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level (J. Math. Soc. Japan; with Pitale (Oklahoma), Saha (QMUL), Schmidt (Oklahoma)); - Products of Eisenstein series and Fourier expansions of modular forms at cusps (Journal of Number theory; with M. Neururer (Darmstadt)). |
| Exploitation Route | The findings are primarily of interest to number theorists and algebraic geometers, especially those working on the arithmetic of curves and abelian varieties, both with theoretical and computational methods. The results have already been applied by the PI and S. Anni (Univ. Heidelberg) to the study of Galois representations: - Constructing hyperelliptic curves with surjective Galois representations (arxiv:1701.05915). The cluster pictures theory has also been pushed further - it can now be used for understanding differentials and local root numbers of hyperelliptic curves, thanks to the works of Kunzweiler (Univ. Ulm) and Bisatt (King's College London, MPIM Bonn and Univ. Brisol): - Differential forms on hyperelliptic curves with semistable reduction (Kunzweiler, arxiv:1902.07784); - Explicit root numbers of abelian varieties (Bisatt; Trans. AMS); - Clusters, inertia, and root numbers (M. Bisatt, arXiv:1902.08981). There is further ongoing work by the PI's group to generalise the techniques to more general settings. |
| Sectors | Other |
| Description | Research Fellows Enhancement Award |
| Amount | £197,953 (GBP) |
| Funding ID | RGF\EA\181052 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 10/2018 |
| End | 03/2021 |
| Description | Selmer groups, arithmetic statistics, and parity conjectures. |
| Amount | £281,350 (GBP) |
| Funding ID | EP/V006541/1 |
| Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
| Sector | Public |
| Country | United Kingdom |
| Start | 10/2021 |
| End | 09/2024 |
| Description | University Research Fellowships Renewals |
| Amount | £281,992 (GBP) |
| Funding ID | URF\R\180011 |
| Organisation | The Royal Society |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 10/2018 |
| End | 09/2021 |