Constructing Galois representations with large Iwasawa $λ$-Invariant
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866911835054145536 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $λ$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for any natural number $n$, one constructs a modular Galois representation such that the associated $λ$-invariant is $\geq n$. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form $f_1$ satisfying suitable conditions, one constructs a congruent modular form $f_2$ for which the $λ$-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_00147 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Constructing Galois representations with large Iwasawa $λ$-Invariant Ray, Anwesh Number Theory 11F80, 11F11, 11R23 Let $p\geq 5$ be a prime. We construct modular Galois representations for which the $\mathbb{Z}_p$-corank of the $p$-primary Selmer group (i.e., $λ$-invariant) over the cyclotomic $\mathbb{Z}_p$-extension is large. More precisely, for any natural number $n$, one constructs a modular Galois representation such that the associated $λ$-invariant is $\geq n$. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form $f_1$ satisfying suitable conditions, one constructs a congruent modular form $f_2$ for which the $λ$-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples. |
| title | Constructing Galois representations with large Iwasawa $λ$-Invariant |
| topic | Number Theory 11F80, 11F11, 11R23 |
| url | https://arxiv.org/abs/2105.00147 |