Modular analogs of character formulas and minimal lifts of modular forms
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910269743038464 |
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| author | Allen, Patrick B. Wake, Preston |
| author_facet | Allen, Patrick B. Wake, Preston |
| contents | If $f$ is a mod-$3$ eigenform of weight 2 and level $Γ_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of $\mathrm{GL}_2(\mathbb{F}_\ell)$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Broué and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of $\mathrm{GL}_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular analogs of character formulas and minimal lifts of modular forms Allen, Patrick B. Wake, Preston Number Theory Representation Theory If $f$ is a mod-$3$ eigenform of weight 2 and level $Γ_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of $\mathrm{GL}_2(\mathbb{F}_\ell)$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Broué and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of $\mathrm{GL}_2$. |
| title | Modular analogs of character formulas and minimal lifts of modular forms |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2509.21426 |