Anticyclotomic Iwasawa main conjectures for modular forms
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912979231965184 |
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| author | Longo, Matteo Pati, Maria Rosaria Vigni, Stefano |
| author_facet | Longo, Matteo Pati, Maria Rosaria Vigni, Stefano |
| contents | Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis relative to $N$. In this paper, we prove (under mild arithmetic assumptions) Iwasawa main conjectures for $f$ over the anticyclotomic $\mathbb Z_p$-extension of $K$ both in the definite setting and in the indefinite setting (in the second case, we prove a main conjecture à la Perrin-Riou for modular forms). Our strategy of proof follows the approach of Bertolini-Darmon via congruences combined with our previous results on an analogue for $f$ of Kolyvagin's conjecture on the non-triviality of his $p$-adic system of derived Heegner points on elliptic curves. As a second contribution, when $p$ splits in $K$ we prove an Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna and Brooks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22483 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anticyclotomic Iwasawa main conjectures for modular forms Longo, Matteo Pati, Maria Rosaria Vigni, Stefano Number Theory Algebraic Geometry 11F11, 11R23 Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis relative to $N$. In this paper, we prove (under mild arithmetic assumptions) Iwasawa main conjectures for $f$ over the anticyclotomic $\mathbb Z_p$-extension of $K$ both in the definite setting and in the indefinite setting (in the second case, we prove a main conjecture à la Perrin-Riou for modular forms). Our strategy of proof follows the approach of Bertolini-Darmon via congruences combined with our previous results on an analogue for $f$ of Kolyvagin's conjecture on the non-triviality of his $p$-adic system of derived Heegner points on elliptic curves. As a second contribution, when $p$ splits in $K$ we prove an Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna and Brooks. |
| title | Anticyclotomic Iwasawa main conjectures for modular forms |
| topic | Number Theory Algebraic Geometry 11F11, 11R23 |
| url | https://arxiv.org/abs/2603.22483 |