Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910868419117056 |
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| author | Mastella, Luca |
| author_facet | Mastella, Luca |
| contents | In this article we prove a refinement of a theorem of Longo and Vigni in the anticyclotomic Iwasawa theory for modular forms. More precisely we give a definition for the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K)$ and $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty)$ of a modular form $f$ of weight $k >2$, over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K) = \widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty) = 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_13134 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory Mastella, Luca Number Theory 11R23 (Primary), 11F11 (Secondary) In this article we prove a refinement of a theorem of Longo and Vigni in the anticyclotomic Iwasawa theory for modular forms. More precisely we give a definition for the ($\mathfrak{p}$-part of the) Shafarevich-Tate groups $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K)$ and $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty)$ of a modular form $f$ of weight $k >2$, over an imaginary quadratic field $K$ satisfying the Heegner hypothesis and over its anticyclotomic $\mathbb{Z}_p$-extension $K_\infty$ and we show that if the basic generalized Heegner cycle $z_{f, K}$ is non-torsion and not divisible by $p$, then $\widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K) = \widetilde{\mathrm{sha}}_{\mathfrak{p}^\infty}(f/K_\infty) = 0$. |
| title | Vanishing of the p-part of the Shafarevich-Tate group of a modular form and its consequences for Anticyclotomic Iwasawa Theory |
| topic | Number Theory 11R23 (Primary), 11F11 (Secondary) |
| url | https://arxiv.org/abs/2307.13134 |