Representation growth of Fuchsian groups and modular forms
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916760690622464 |
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| author | Larsen, Michael Taylor, Jay Tiep, Pham Huu |
| author_facet | Larsen, Michael Taylor, Jay Tiep, Pham Huu |
| contents | Let $Γ$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $Γ$. Then $|\mathrm{Hom}(Γ,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-χ(Γ))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation growth of Fuchsian groups and modular forms Larsen, Michael Taylor, Jay Tiep, Pham Huu Group Theory Number Theory 20H10 (Primary) 11F20, 11F27, 20C15, 20C33, 20G40 (Secondary) Let $Γ$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $Γ$. Then $|\mathrm{Hom}(Γ,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-χ(Γ))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight. |
| title | Representation growth of Fuchsian groups and modular forms |
| topic | Group Theory Number Theory 20H10 (Primary) 11F20, 11F27, 20C15, 20C33, 20G40 (Secondary) |
| url | https://arxiv.org/abs/2407.07193 |