Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914983847133184 |
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| author | Kumar, Arvind Mishra, Prabhat Kumar |
| author_facet | Kumar, Arvind Mishra, Prabhat Kumar |
| contents | Let $k \ge 2$ be an even integer, $ \ell \ge \max\{5, k-1\} $ be a prime, and $N$ be a squarefree positive integer. It is known that if the $\rm{mod}\,\ell$ Galois representation $\overlineρ_f$ associated with a newform $f$ of weight $k$, level $N$, and trivial nebentypus is reducible, then $\overlineρ_f \simeq 1 \oplus \overlineχ_\ell^{k-1}$, up to semisimplification, where $\overlineχ_\ell^{}$ is the $\rm{mod}\,\ell$ cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$, level $N$ with exactly two prime factors with specified Atkin-Lehner eigenvalues. Specifically, this proves a conjecture of Billerey and Menares when $N$ is a product of two primes under some mild assumption. As an application, we show that for any $\ell\ge 5$ and $k=2$ or $\ell+1$, there exist a large class of distinct primes $p$ and $q$ such that the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$ and level $pq$ with explicit Atkin-Lehner eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations Kumar, Arvind Mishra, Prabhat Kumar Number Theory 11F33, 11F11, 11F80, 11N37 Let $k \ge 2$ be an even integer, $ \ell \ge \max\{5, k-1\} $ be a prime, and $N$ be a squarefree positive integer. It is known that if the $\rm{mod}\,\ell$ Galois representation $\overlineρ_f$ associated with a newform $f$ of weight $k$, level $N$, and trivial nebentypus is reducible, then $\overlineρ_f \simeq 1 \oplus \overlineχ_\ell^{k-1}$, up to semisimplification, where $\overlineχ_\ell^{}$ is the $\rm{mod}\,\ell$ cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$, level $N$ with exactly two prime factors with specified Atkin-Lehner eigenvalues. Specifically, this proves a conjecture of Billerey and Menares when $N$ is a product of two primes under some mild assumption. As an application, we show that for any $\ell\ge 5$ and $k=2$ or $\ell+1$, there exist a large class of distinct primes $p$ and $q$ such that the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$ and level $pq$ with explicit Atkin-Lehner eigenvalues. |
| title | Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations |
| topic | Number Theory 11F33, 11F11, 11F80, 11N37 |
| url | https://arxiv.org/abs/2410.16854 |