Hyperelliptic curves of genus three over finite fields of even characteristic

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic. Also, we compute the automorphism group of each curve and an explicit formula for the total number of curves.     

Teichmüller Curves Through Fermat Curves

The billiard in a regular n-gon is known to give rise to a Teichmüller curve. For odd n, we compute the genus of this curve, a number field over which the curve may be defined and branched covering relations between certain pairs of these curves. If n is a power of a prime congruent to 3 or 5 modulo 8, the Teichmüller curve may be defined over the rationals. Received: June 2006, Revision: October 2006, Accepted: November 2006     

Non-constant curves of genus 2 with infinite pro-Galois covers

For every odd prime number p, we give examples of non-constant smooth families of curves of genus 2 over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.     

On certain curves of genus three in characteristic two

We study curves of genus 3 over algebraically closed fields of characteristic 2 with the canonical theta characteristic totally supported in one point. We compute the moduli dimension of such curves and focus on some of them which have two Weierstrass points with Weierstrass directions towards the support of the theta characteristic. We answer questions related to order sequence and Weierstrass weight of Weierstrass points and the existence of other Weierstrass points with similar properties. – Dedicated to the treasured memory of our coauthor, Paulo Henrique Viana     

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus of the moduli space M₃ of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385-389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198-200] on the hyperelliptic locus we obtain a complete picture of these strata for M₃.     

Fields of moduli and definition of hyperelliptic curves of odd genus

Mestre has shown that if a hyperelliptic curve C of even genus is defined over a subfield k ì \mathbbC{k \subset \mathbb{C}} then C can be hyperelliptically defined over the same field k. In this paper, for all genera g > 1, g o 1{g\equiv1} mod 4, hence odd, we construct an explicit hyperelliptic curve defined over \mathbbQ{\mathbb{Q}} which can not be hyperelliptically defined over \mathbbQ{\mathbb{Q}}.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

Isomorphism classes of elliptic and hyperelliptic curves over finite fields

We give the number and representatives of isomorphism classes of hyperelliptic curves of genus g defined over finite fields , g=1,2,3. These results have applications to hyperelliptic curve cryptography.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

The modularity of some Q-curves

A Q-curve is an elliptic curve, defined over a number field, that is isogenous to each of its Galois conjugates. Ribet showed that Serre's conjectures imply that such curves should be modular. Let E be an elliptic curve defined over a quadratic field such that E is 3-isogenous to its Galois conjugate. We give an algorithm for proving any such E is modular and give an explicit example involving a quotient of J_o (169). As a by-product, we obtain a pair of 19-isogenous elliptic curves, and relate this to the existence of a rational point of order 19 on J₁ (13).     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY ²¹ =bX ²¹ +cZ ²¹ defined over finite fields F_q such thatq = p ^α? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_q. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_q, as rational functions in the variablet, for distinct cases ofa, b, andc, in F _q ^* . Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_q. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).     

On the structure of generalized symmetric spaces of SLn𝔽q)

In this paper we extend previous results regarding SL₂(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SL_n(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SL_n(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.     

Halving for the 2-Sylow subgroup of genus 2 curves over binary fields

We give a deterministic polynomial time algorithm to find the structure of the 2-Sylow subgroup of the Jacobian of a genus 2 curve over a finite field of characteristic 2. Our procedure starts with the points of order 2 and then performs a chain of successive halvings while such an operation makes sense. The stopping condition is triggered when certain polynomials fail to have roots in the base field, as previously shown by I. Kitamura, M. Katagi and T. Takagi. The structure of our algorithm is similar to the already known case of genus 1 and odd characteristic.     

k-Involutions of SL(n,k) over fields of characteristic 2

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n,k) in the case that k is any field of characteristic 2; we also describe fixed-point groups of each type of involution.     

Enumerative Geometry for Plane Cubic Curves in Characteristic 2

Consider the plane cubic curves over an algebraically closed field of characteristic 2. By blowing up the parameter space P⁹ twice we obtain a variety B of complete cubics. We then compute the characteristic numbers for various families of cubics by intersecting cycles on B.     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

Non-hyperelliptic modular curves of genus 3

A curve C defined over Q is modular of level N if there exists a non-constant morphism from X₁(N) onto C defined over Q for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is Q-isogenous to a given three dimensional Q-quotient of J₁(N). Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J₁(N) and Jac C factors through the new part of J₁(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves and we present some examples that show the ambiguity of the non-new modular case.     

The Chen–Ruan cohomology of moduli of curves of genus 2 with marked points

In this work we describe the Chen–Ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points. In the first half of the paper we compute the additive structure of the Chen–Ruan cohomology ring for the moduli stack of stable n-pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen–Ruan cohomology ring as an algebra on the ordinary cohomology.