Related papers: On Shimura curves in the Schottky locus

We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.

Craw and Ishii proved that for a finite abelian group G in SL_3(C) every (projective) relative minimal model of C^3/G is isomorphic to the fine moduli space of \theta-stable G-constellations for some GIT parameter \theta. In this article,…

We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$. There is a homomorphism between the corresponding Hurwitz space $\H$ of such covers to the moduli space $\M_g$ of genus $g$ curves. In the case $g=3$,…

McMullen's compact Kobayashi-geodesic curve $V \subset X_L$, arising from the hyperbolic triangle group $\Delta(14,21,42)$ via a modular embedding into the Hilbert modular sixfold $X_L = \mathbb{H}^6/\mathrm{SL}_2(\mathcal{O}_L)$ attached…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured…

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with…

This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a…

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a…

In this paper we classify isogeny classes of global $\mathsf{G}$-shtukas over a smooth projective curve $C/\mathbb{F}_q$ (or equivalently $\sigma$-conjugacy classes in $\mathsf{G}(\mathsf{F} \otimes_{\mathbb{F}_q} \overline{\mathbb{F}_q})$…

We prove the non-existence of Shimura subvarieties of positive dimension contained generically in the hyperelliptic Torelli locus for curves of genus at least 8, which is an analogue of Oort's conjecture in the hyperelliptic case.

For a point $x_0$ in a Shimura variety attached to a Shimura datum of Hodge type $(G,X)$, we have an associated abelian scheme $A_0$. Fixing a non-empty finite set $\mathcal{S}$ of primes, we consider the simultaneous supersingular…

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the…

We prove an hyperbolic analogue of the Bloch-Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. We consider the case of non compact Shimura varieties completing the proof of the result for all Shimura…

We study metric measure spaces that admit "thick" families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into…

In this paper we characterize the genera of those quotient curves $\mathcal{H}_q/G$ of the $\mathbb{F}_{q^2}$-maximal Hermitian curve $\mathcal{H}_q$ for which $G$ is contained in the maximal subgroup $\mathcal{M}_1$ of ${\rm…

We study wildly ramified G-Galois covers $\phi:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia…

A conjecture in [Ish20] states that for a finite subgroup $G$ of $GL(2; \mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability parameter…

Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces…

Let $X$ be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli space of rank-2 bundles. We show that up to isomorphism, there is only one (up to…