Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$
We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001336/type/journal_article |
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| author | Timo Keller Michael Stoll |
| author_facet | Timo Keller Michael Stoll |
| author_sort | Timo Keller |
| collection | DOAJ |
| description | We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over
${\mathbf {Q}}$
. We apply our methods to all 28 Atkin–Lehner quotients of
$X_0(N)$
of genus
$2$
, all 97 genus
$2$
curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least
$2$
. We also give an example where we verify that the order of the Tate–Shafarevich group is
$7^2$
and agrees with the order predicted by the BSD Conjecture. |
| format | Article |
| id | doaj-art-6c643502fe404b9ca6eb1bf0502be8af |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-6c643502fe404b9ca6eb1bf0502be8af2025-01-30T05:12:56ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.133Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$Timo Keller0https://orcid.org/0000-0003-0916-8478Michael Stoll1https://orcid.org/0000-0001-5868-2962Rijksuniversiteit Groningen, Bernoulli Institute, Bernoulliborg, Nijenborgh 9, 9747 AG Groningen, The NetherlandsDepartment of Mathematics, Chair of Computer Algebra, Universität Bayreuth, Universitätsstrasse 30, Bayreuth, 95447, Germany; E-mail:We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$ . We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$ , all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$ . We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.https://www.cambridge.org/core/product/identifier/S2050509424001336/type/journal_article11G4011G1014G05 |
| spellingShingle | Timo Keller Michael Stoll Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ Forum of Mathematics, Sigma 11G40 11G10 14G05 |
| title | Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ |
| title_full | Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ |
| title_fullStr | Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ |
| title_full_unstemmed | Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ |
| title_short | Complete verification of strong BSD for many modular abelian surfaces over ${\mathbf {Q}}$ |
| title_sort | complete verification of strong bsd for many modular abelian surfaces over mathbf q |
| topic | 11G40 11G10 14G05 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001336/type/journal_article |
| work_keys_str_mv | AT timokeller completeverificationofstrongbsdformanymodularabeliansurfacesovermathbfq AT michaelstoll completeverificationofstrongbsdformanymodularabeliansurfacesovermathbfq |