Estimates and asymptotics of Teichmüller modular forms
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arXiv
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2024
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| author | Aryasomayajula, Anilatmaja Sadhukhan, Debasish |
| author_facet | Aryasomayajula, Anilatmaja Sadhukhan, Debasish |
| contents | In this article, we derive estimates of Teichmüller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the Deligne-Mumford compactification of $\mathcal{M}_{g}$, and we denote its boundary by $\partial\mathcal{M}_{g}$. Let $π:\mathcal{C}_{g}\longrightarrow\mathcal{M}_{g}$ be the universal surface. For any $n\geq 1$, let $Λ_{n}:=π_{\ast}(T_{v}\mathcal{C}_{g})^{n}$, where $T_{v}\mathcal{C}_{g}$ denotes the vertical holomorphic tangent bundle of the fibration $π$, and the fiber of $Λ_{n}$ over any $X\in\mathcal{M}_{g}$ is equal to $H^{0}(X,Ω_{X}^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $λ_{n}:=\mathrm{det}(Λ_{n})$ denote the determinant line bundle of the vector bundle $Λ_{n}$, whose sections are known as Teichmüller modular forms. The complex vector space of Teichmüller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{\mathrm{Qu}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimates and asymptotics of Teichmüller modular forms Aryasomayajula, Anilatmaja Sadhukhan, Debasish Complex Variables Number Theory 32A10, 30F10 In this article, we derive estimates of Teichmüller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the Deligne-Mumford compactification of $\mathcal{M}_{g}$, and we denote its boundary by $\partial\mathcal{M}_{g}$. Let $π:\mathcal{C}_{g}\longrightarrow\mathcal{M}_{g}$ be the universal surface. For any $n\geq 1$, let $Λ_{n}:=π_{\ast}(T_{v}\mathcal{C}_{g})^{n}$, where $T_{v}\mathcal{C}_{g}$ denotes the vertical holomorphic tangent bundle of the fibration $π$, and the fiber of $Λ_{n}$ over any $X\in\mathcal{M}_{g}$ is equal to $H^{0}(X,Ω_{X}^{\otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $λ_{n}:=\mathrm{det}(Λ_{n})$ denote the determinant line bundle of the vector bundle $Λ_{n}$, whose sections are known as Teichmüller modular forms. The complex vector space of Teichmüller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{\mathrm{Qu}}$. |
| title | Estimates and asymptotics of Teichmüller modular forms |
| topic | Complex Variables Number Theory 32A10, 30F10 |
| url | https://arxiv.org/abs/2412.13997 |