In this article, we derive estimates of Teichm\"uller 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 $\pi:\mathcal{ C}_{ g}\longrightarrow\mathcal{ M}_{ g}$ be the universal surface. For any $n\geq 1$, let $\Lambda_{ n}:=\pi_{ \ast}(T_{ v}\mathcal{ C}_{ g})^{ n}$, where $T_{ v}\mathcal{ C}_{ g}$ denotes the vertical holomorphic tangent bundle of the fibration $\pi$, and the fiber of $\Lambda_{ n}$ over any $X\in\mathcal{ M}_{ g}$ is equal to $H^{ 0}(X,\Omega_{ X}^{ \otimes n})$, the space of holomorphic differentials of degree-$n$, defined over the Riemann surface $X$. Let $\lambda_{ n}:=\mathrm{ det}(\Lambda_{ n})$ denote the determinant line bundle of the vector bundle $\Lambda_{ n}$, whose sections are known as Teichm\"uller modular forms. The complex vector space of Teichm\"uller modular forms is equipped with Quillen metric, which is denoted by $\|\cdot\|_{ \mathrm{ Qu}}$.
Categories: math.CV math.NT