Tamagawa numbers and positive rank of elliptic curves
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918461443145728 |
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| author | Aylward, Edwina |
| author_facet | Aylward, Edwina |
| contents | This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema, and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tamagawa numbers and positive rank of elliptic curves Aylward, Edwina Number Theory This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema, and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers. |
| title | Tamagawa numbers and positive rank of elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.17962 |