Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911375430778880 |
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| author | Fong, Ho Leung |
| author_facet | Fong, Ho Leung |
| contents | In this paper, we relate $L(1,π,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,π,\mathrm{Ad}^\circ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$ Fong, Ho Leung Number Theory 11F67 (Primary) 11F75, 11F33 (Secondary) In this paper, we relate $L(1,π,\mathrm{Ad}^\circ)$ to the congruence ideals for cohomological cuspidal automorphic representations $π$ of $\mathrm{GL}_n$ over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint $L$-functions. For CM fields, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of $L(1,π,\mathrm{Ad}^\circ)$. |
| title | Adjoint $L$-functions, congruence ideals, and Selmer groups over $\mathrm{GL}_n$ |
| topic | Number Theory 11F67 (Primary) 11F75, 11F33 (Secondary) |
| url | https://arxiv.org/abs/2409.14933 |