Diagonal cycles and anticyclotomic Iwasawa theory of modular forms
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arXiv
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| Format: | Preprint |
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2023
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| author | Castella, Francesc Do, Kim Tuan |
| author_facet | Castella, Francesc Do, Kim Tuan |
| contents | We construct a new Euler system for the Galois representation $V_{f,χ}$ attached to a newform $f$ of weight $2r\geq 2$ twisted by an anticyclotomic Hecke character $χ$. The Euler system is anticyclotomic in the sense of Jetchev-Nekovar-Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch-Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa-Greenberg main conjecture for $V_{f,χ}$.
In particular, in the case where the base-change of $f$ to our imaginary quadratic field has root number $+1$ and $χ$ has higher weight (which implies that the complex $L$-function $L(V_{f,χ},s)$ vanishes at the center), our results show that the Bloch-Kato Selmer group of $V_{f,χ}$ is nonzero, as predicted by the Bloch-Kato conjecture; and if in addition a certain distinguished class $κ_{\,f,χ}$ is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch-Kato conjecture for $V_{f,χ}$ were left wide open by the earlier approaches using Heegner cycles and/or Beilinson-Flach elements. Our construction is based instead on a generalization of the Gross-Kudla-Schoen diagonal cycles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_06751 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Diagonal cycles and anticyclotomic Iwasawa theory of modular forms Castella, Francesc Do, Kim Tuan Number Theory We construct a new Euler system for the Galois representation $V_{f,χ}$ attached to a newform $f$ of weight $2r\geq 2$ twisted by an anticyclotomic Hecke character $χ$. The Euler system is anticyclotomic in the sense of Jetchev-Nekovar-Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch-Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa-Greenberg main conjecture for $V_{f,χ}$. In particular, in the case where the base-change of $f$ to our imaginary quadratic field has root number $+1$ and $χ$ has higher weight (which implies that the complex $L$-function $L(V_{f,χ},s)$ vanishes at the center), our results show that the Bloch-Kato Selmer group of $V_{f,χ}$ is nonzero, as predicted by the Bloch-Kato conjecture; and if in addition a certain distinguished class $κ_{\,f,χ}$ is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch-Kato conjecture for $V_{f,χ}$ were left wide open by the earlier approaches using Heegner cycles and/or Beilinson-Flach elements. Our construction is based instead on a generalization of the Gross-Kudla-Schoen diagonal cycles. |
| title | Diagonal cycles and anticyclotomic Iwasawa theory of modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.06751 |