We prove that for any pair of integers 0\leq r\leq g such that g\geq 3 or r>0, there exists a (hyper)elliptic curve C over F_2 of genus g and 2-rank r whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties (A,λ) over F_2 of dimension g and 2-rank r such that \Aut(A,λ)={\pm 1}.
This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring-Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic $p>2$.
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists such a curve of genus $g$ in general characteristic $p > 0$, and in the case of $g=4$, several computational approaches to search for those curves have been proposed. In the genus-$4$ hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp.\ curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case w…
In this paper, we list all cyclic automorphisms subgroups $H$ for which there exists a smooth projective non-hyperelliptic sextic curve $C$ with $H\preceq Aut(C)$. Furthermore, we attach to each group a defining equation of a plane sextic curve having exactly this group as cyclic automorphisms subgroup. These equations cover up to isomorphism all plane non-singular sextic curves having some non-trivial automorphism.
For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.
Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σthat fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with σbeing the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms σ' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with σ' being its hyperelliptic involution.
Let alpha be an automorphism of a hyperelliptic curve C of genus g, and let alpha' be the automorphism of P^1 induced by alpha. Let n be the order of alpha and let n' be the order of alpha'. We show that the triple (g,n,n') completely determines the characteristic polynomial of the automorphism alpha^* of the Jacobian of C, unless n is even, n=n', and (2g+2)/n is even, in which case there are two possibilities. We give explicit formulas for the characteristic polynomial in all cases.
This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\math…
Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.
In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2 (k)$-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus. The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decompo…
We determine all modular curves X(N) (with $N\geq 7$) which are hyperelliptic or bielliptic. We make available a proof that the automorphism group of X(N) coincides with the normalizer of $Γ(N)$ in $\operatorname{PSL}_2(\mathbb{R})$.
We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.
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 curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model $y^2=F(x)$ or $y^2=x F(x)$ of the curve over its field of moduli where $F(x)$ can be chosen to be decomposable in $x^2$ or $x^5$. While similar equations have been given in Bujalance, Cirre, Gamboa and Gr…
We describe an algorithm to compute the zeta function of any non-hyperelliptic genus 3 plane curve $C$ over a finite field with automorphism group $G = \mathbb{Z} / 2 \mathbb{Z}$. This algorithm computes in the Monsky-Washnitzer cohomology of~the curve. Using the relation between the Monsky-Washnitzer cohomology of $C$ and its quotient $E := C/G$, the computation splits into 2 parts: one in a subspace of the Monsky-Washnitzer cohomology and a second which reduces to the point counting on an elliptic curve $E$. The former corresponds to the dimension $2$ abelian surface $\mathrm{ker}(\mathrm{Ja…
We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the reduced automorphism group of $\p$ has at least two involutions, there exists a representative $C$ of the isomorphism class $\p$ which is defined over $L$.
An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that …
A group $H \cong {\mathbb Z}_{2}^{n}$, $n \geq 3$, of conformal automorphisms of a closed Riemann surface $S$ such that $S/H$ has genus zero and exactly $(n+1)$ cone points is called a generalized Humbert group of type $n$, in which case, $S$ is called a generalized Humbert curve of type $n$. It is known that a generalized Humbert curve $S$ of type $n \geq 4$ is non-hyperelliptic and that it admits a unique generalized Humbert group $H$ of type $n$. We describe those subgroups $K$ of $H$, acting freely on $S$, such that $S/K$ is hyperelliptic.
Classical modular curves are of deep interest in arithmetic geometry. In this survey we show how the work of Fumiyuki Momose is involved in order to list the classical modular curves which satisfy that the set of quadratic points over $\mathbb{Q}$ is infinite. In particular we recall results of Momose on hyperelliptic modular curves and on automorphisms groups of modular curves. Moreover, we fix some inaccuracies of the existing literature in few statements concerning automorphism groups of modular curves and we make available different results that are difficult to find a precise reference, f…
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curv…
We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve.
Let X be a smooth projective hyperelliptic curve over an algeraically closed field k of prime characteristic p. The aim of this note is to find necessary and sufficient conditions on the automorphism group of the curve X to be lifted to characteristic zero. The results will be generalised for a certain family of curves that we call cyclic curves.
We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.
Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic non-trigonal, whose model is a complete intersection of three quadrics in $\mathbb{P}^4$. We present and explain algorithms we used to determine, up to isomorphism over $\mathbb{F}_2$, all genus 5 curves defined over $\mathbb{F}_2$, and we do that separately for each of the three mentioned types. We consider these curves in terms of isogeny classes over $\mathbb{F}_2$ of their Jacobians or their Newton polygons, and for each of the three types, we compute the number of curves over $\mathbb{F}_2$ weighted by the size of their $…
In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $§$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $§$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in §$…
Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $π: \X \longrightarrow ¶^1,$ where $¶^1$ denotes the projective line. The problem of expressing branch points of the covering $π$ in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among…
The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $Ł_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a birational parametrization of $Ł_g$ via dihedral invariants and show how these invariants can be used to determine the field of moduli of points $\p \in Ł_g$. We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli is a field of definition and prove this conjecture for any point $\p\in Ł_g$ such that the Klein 4-group is embedded in the red…
Let $p$ be a large prime, and let $C$ be a hyperelliptic curve over $\mathbb{F}_p$. We study the distribution of the $x$-coordinates in short intervals when the $y$-coordinates lie in a prescribed interval, and the distribution of the distance between consecutive $x$-coordinates with the same property. Next, let $g(P,P_0)$ be a rational function of two points on $C$. We study the distribution of the above distances with an extra condition that $g(P_i,P_{i+1})$ lies in a prescribed interval, for any consecutive points $P_i,P_{i+1}$.
In this paper we study the automorphism groups of real curves admitting a regular meromorphic function $f$ of degree $p$, so called real cyclic $p$-gonal curves. When $p=2$ the automorphism groups of real hyperelliptic curves where given by Bujalance et al. in \cite{BCGG}.
We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the second part of the paper, we show that finite abelian level covers of the hyperelliptic locus …
We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that o…
We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\mathcal X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$ decomposes.
Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of t…
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then character…
Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$. If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it i…
In this paper we study rank two commuting ordinary differential operators with polynomial coefficients and the orbit space of the automorphisms group of the first Weyl algebra on such operators. We prove that for arbitrary fixed spectral curve of genus one the space of orbits is infinite. Moreover, we prove in this case that for for any $n\ge 1$ there is a pair of self-adjoint commuting ordinary differential operators of rank two $L_4=(\partial_x^2+V(x))^2+W(x)$, $L_{6}$, where $W(x),V(x)$ are polynomials of degree $n$ and $n+2$. We also prove that there are hyperelliptic spectral curves with …
Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over $k$ for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the al…
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
We propose an object called 'sepcanonical system' on a stable curve $X_0$ which is to serve as limiting object- distinct from other such limits introduced previously- for the canonical system, as a smooth curve degenerates to $X_0$. First for curves which cannot be separated by 2 or fewer nodes, the so-called '2-inseparable' curves, the sepcanonical system is just the sections of the dualizing sheaf, which is not very ample iff $X_0$ is a limit of smooth hyperelliptic curves (such $X_0$ are called 2-inseparable hyperelliptics). For general, 2-separable curves $X_0$ this assertion is false, lea…
It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting to know, in terms of the algebraic structure of $G$, if $S$ can in fact be defined over ${\mathbb Q}$. This is the situation if $G$ is either abelian or isomorphic to $A \rtimes {\mathbb Z}_{2}$, where $A$ is an abelian group. On the other hand, as shown by Streit and Wolfart, if $G = {\mathbb Z}_{p} \rtimes {\mathbb Z}_{q}$ where $p,q>3$ are prime integers,…
In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. In particular, our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.
Let $C_k$ be a smooth projective curve over a global field $k$, which is neither rational nor elliptic. Harris-Silverman, when $p=0$, and Schweizer, when $p>0$ together with an extra condition on the Jacobian variety $\operatorname{Jac}(C_k)$ arising from Mordell's conjecture, showed that $C$ has infinitely many quadratic points over some finite field extension $L/k$ inside $\overline{k}$ (a fixed algebraic closure of $k$) if and only if $C$ is hyperelliptic or bielliptic. Now, let $C_k$ be a smooth plane curve of a fixed degree $d\geq4$ with $p=0$ or $p>(d-1)(d-2)+1$ (up to an extra condition…
We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.
We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definiti…
