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Let $X(2)$ denote the compact Riemann surface obtained by compactifying $Y(2) = \Gamma(2)\mathfrak{h}$ by adding cusps. The modular $\lambda$-function on the complex upper half plane $\mathfrak{h}$ …

Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^ …

Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero. Let $\mathcal C$ be a class of curve …

Recently, in my research I bumped onto gonal morphisms. At the moment, my knowledge is based upon some things I read on the internet. Before stating my questions, I added some definitions/facts that m …

Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways. For example, one could fix the number $r$ of branch points, the degree $n$ of the cover a …

I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two. Let $X$ …

I have the following question. It's a long shot, but worth the try. Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ suc …

Let $C$ be a smooth projective connected curve of genus $g$ over $\bar{\mathbf{Q}}$. Fix a finite non-empty (Edit) set of closed points $S$ in $C$ and let $U$ be the complement of $S$ in $C$. Q1. (Al …