Related papers: Supersingular genus-two curves over fields of char…

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…

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental…

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\ell}: X^nY^\ell + Y^nZ^\ell + Z^nX^\ell = 0$, with $n$ and $\ell$ relatively prime, is supersingular over the finite field $\mathbb{F}_{p}$ if and…

In this first of a series of three papers we outline an approach to classifying 4d $\mathcal{N}{=}2$ superconformal field theories at rank 2. The classification of allowed scale invariant $\mathcal{N}=2$ Coulomb branch geometries of…

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic…

The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the…

We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all…

Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g_1,...,g_n of degree d in B=k[x,y] which parameterize C in a birational, base point free, manner. We study the…

We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian…

A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate…

Using Galois-Stiefel-Whitney classes of theta characteristics we show that over a totally real base field the moduli stack of smooth genus $g$ curves and the moduli stack of principally polarized abelian varieties of dimension $g$ have…

Let C be a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation defining C up to isomorphism. In this paper we consider genus one equations of degree n defining C, namely a…

Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to…

We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular curve…

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the…

Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.

In this article we consider moduli properties of singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3 surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$ be the stack…

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by…

In "Proving that a genus 2 curve has complex multiplication", van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over…

We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians…