Mazur's growth number conjecture and congruences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908380026634240 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under $p$-congruences between Galois representations. As a first step, I establish Mazur's conjecture for certain triples $(E, K, p)$ under explicit hypotheses. Building on this, I prove analogous results for Greenberg Selmer groups attached to modular forms that are congruent mod $p$ to $E$, including all specializations arising from Hida families of fixed tame level. In particular, I show that the Mordell-Weil ranks in non-anticyclotomic $\mathbb{Z}_p$-extensions of $K$ remain bounded for elliptic curves $E'$ such that $E[p]$ and $E'[p]$ are isomorphic as Galois modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mazur's growth number conjecture and congruences Ray, Anwesh Number Theory 11R23, 11G05 Motivated by the work of Greenberg-Vatsal and Emerton-Pollack-Weston, I investigate the extent to which Mazur's conjecture on the growth of Selmer ranks in $\mathbb{Z}_p$-extensions of an imaginary quadratic field persists under $p$-congruences between Galois representations. As a first step, I establish Mazur's conjecture for certain triples $(E, K, p)$ under explicit hypotheses. Building on this, I prove analogous results for Greenberg Selmer groups attached to modular forms that are congruent mod $p$ to $E$, including all specializations arising from Hida families of fixed tame level. In particular, I show that the Mordell-Weil ranks in non-anticyclotomic $\mathbb{Z}_p$-extensions of $K$ remain bounded for elliptic curves $E'$ such that $E[p]$ and $E'[p]$ are isomorphic as Galois modules. |
| title | Mazur's growth number conjecture and congruences |
| topic | Number Theory 11R23, 11G05 |
| url | https://arxiv.org/abs/2505.19542 |