Let $X$ and $X^{\prime }$ be compact Riemann surfaces of genus $\ge 3$, and let $G$ and $G^{\prime }$ be nonabelian reductive complex groups. If one component ${ℳ}_G^d\left(X\right)$ of the coarse moduli space for semistable principal $G$–bundles over $X$ is isomorphic to another component ${ℳ}_{G^{\prime }}^{d^{\prime }}\left(X^{\prime }\right)$, then $X$ is isomorphic to $X^{\prime }$.

Let $X$ be a general proper and smooth curve of genus $2$ (resp. of genus $3$) defined over an algebraically closed field of characteristic $p$. When $3\le p\le 7$, the action of Frobenius on rank $2$ semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order $2$ line bundle over $X$. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp....

In this note we show that, for any log-canonical pair $(X,\Delta )$, $K_X+\Delta$ is $\mathbb{Q}$-effective if its Chern class contains an effective $\mathbb{Q}$-divisor. Then, we derive some direct corollaries.

Let $𝒳\to S$ be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and $ℱ$ a $G$-bundle over $𝒳$ with connection along the fibres $𝒳\to S$. We construct a line bundle with connection $({ℒ}_{ℱ},{\nabla }_{ℱ})$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $({ℒ}_{ℱ},{\nabla }_{ℱ})$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X\times S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $({ℒ}_{ℱ},{\nabla }_{ℱ})$ provides a prequantization...

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set $${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$ Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series...