The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929604298539008 |
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| author | Paterson, Ross |
| author_facet | Paterson, Ross |
| contents | We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups of E/K is bounded. In particular, this provides a bound for the average K-rank of elliptic curves E/Q for such K. Additionally, we give bounds for certain representation-theoretic invariants of Mordell--Weil groups over Galois extensions of such F.
The central result is: for each finite Galois extension K/F of number fields and prime number p, as E/Q varies, the difference in dimension between the Galois fixed space in the p-Selmer group of E/K and the p-Selmer group of E/F has bounded average. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_02486 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves Paterson, Ross Number Theory 11G05 (Primary) 11G07, 11N45, 14H52 (Secondary) We show that if F is the rational numbers or a multiquadratic number field, p is 2,3, or 5, and K/F is a Galois extension of degree a power of p, then for elliptic curves E/Q ordered by height, the average dimension of the p-Selmer groups of E/K is bounded. In particular, this provides a bound for the average K-rank of elliptic curves E/Q for such K. Additionally, we give bounds for certain representation-theoretic invariants of Mordell--Weil groups over Galois extensions of such F. The central result is: for each finite Galois extension K/F of number fields and prime number p, as E/Q varies, the difference in dimension between the Galois fixed space in the p-Selmer group of E/K and the p-Selmer group of E/F has bounded average. |
| title | The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves |
| topic | Number Theory 11G05 (Primary) 11G07, 11N45, 14H52 (Secondary) |
| url | https://arxiv.org/abs/2106.02486 |