The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918494918934528 |
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| author | Dummigan, Neil Golyshev, Vasily de Jeu, Rob Kerr, Matt |
| author_facet | Dummigan, Neil Golyshev, Vasily de Jeu, Rob Kerr, Matt |
| contents | For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves Dummigan, Neil Golyshev, Vasily de Jeu, Rob Kerr, Matt Number Theory Algebraic Geometry K-Theory and Homology 11G40, 19F27 For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture. |
| title | The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves |
| topic | Number Theory Algebraic Geometry K-Theory and Homology 11G40, 19F27 |
| url | https://arxiv.org/abs/2605.11100 |