Iwasawa invariants of modular forms with $a_p=0$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915577623216128 |
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| author | Gajek-Leonard, Rylan |
| author_facet | Gajek-Leonard, Rylan |
| contents | Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $θ_n(f)$, both of which encode arithmetic data about $f$ along the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$. We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed $p$-adic $L$-functions. As corollaries, we determine the $p$-adic valuation of critical values of the $L$-function of $f$, and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and $p+1$. Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight $k\geq 2$ with $a_p=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05748 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Iwasawa invariants of modular forms with $a_p=0$ Gajek-Leonard, Rylan Number Theory Primary 11R23, Secondary 11F11, 11F33 Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $θ_n(f)$, both of which encode arithmetic data about $f$ along the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$. We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed $p$-adic $L$-functions. As corollaries, we determine the $p$-adic valuation of critical values of the $L$-function of $f$, and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and $p+1$. Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight $k\geq 2$ with $a_p=0$. |
| title | Iwasawa invariants of modular forms with $a_p=0$ |
| topic | Number Theory Primary 11R23, Secondary 11F11, 11F33 |
| url | https://arxiv.org/abs/2302.05748 |