Deep congruences + the Brauer-Nesbitt theorem
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913585574182912 |
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| author | Anni, Samuele Ghitza, Alexandru Medvedovsky, Anna |
| author_facet | Anni, Samuele Ghitza, Alexandru Medvedovsky, Anna |
| contents | We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo divided-power ideals. Our approach is combinatorial: we introduce a $p$-equivalence relation on partitions, and use it to prove that certain linear combinations of power-sum functions are $p$-integral. We also include a second proof, short and algebraic, suggested by an anonymous referee. As a corollary we obtain a refinement of the Brauer-Nesbitt theorem for a single linear operator, motivated by the study of Hecke modules of mod-$p$ modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_07108 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deep congruences + the Brauer-Nesbitt theorem Anni, Samuele Ghitza, Alexandru Medvedovsky, Anna Combinatorics Commutative Algebra Number Theory 05E05 (Primary) 05E40, 05E10, 11F33, 11P83, 11T99, 20C99 (Secondary) We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo divided-power ideals. Our approach is combinatorial: we introduce a $p$-equivalence relation on partitions, and use it to prove that certain linear combinations of power-sum functions are $p$-integral. We also include a second proof, short and algebraic, suggested by an anonymous referee. As a corollary we obtain a refinement of the Brauer-Nesbitt theorem for a single linear operator, motivated by the study of Hecke modules of mod-$p$ modular forms. |
| title | Deep congruences + the Brauer-Nesbitt theorem |
| topic | Combinatorics Commutative Algebra Number Theory 05E05 (Primary) 05E40, 05E10, 11F33, 11P83, 11T99, 20C99 (Secondary) |
| url | https://arxiv.org/abs/2207.07108 |