Hurwitz Curves Research Articles

Systolic length of triangular modular curves

We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.

The supersingularity of Hurwitz curves

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 only if there exists an integer $i$ such that $p^i \equiv -1 \bmod (n^2 - n\ell + \ell^2)$. If this holds, we prove that it is also true that the curve is maximal over $\mathbb{F}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.

About the Fricke–Macbeath curve

A closed Riemann surface of genus $$g \geqslant 2$$ is called a Hurwitz curve if its group of conformal automorphisms has order $$84(g-1)$$ . In 1895, Wiman noticed that there is no Hurwitz curve of genus $$g=2,4,5,6$$ and, up to isomorphisms, there is a unique Hurwitz curve of genus $$g=3$$ ; this being Klein’s plane quartic curve. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $$g=7$$ ; this known as the Fricke–Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces $$S_{\mu }$$ admitting a group of conformal automorphisms so that $$S_{\mu }/G_{\mu }$$ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath’s description. We also observe that the jacobian variety of each $$S_{\mu }$$ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke–Macbeath curve, we obtain the well-known fact that its jacobian variety is isogenous to $$E^{7}$$ for a suitable elliptic curve E.

The a-numbers of Fermat and Hurwitz curves

For an algebraic curve X defined over an algebraically closed field of characteristic p>0, the a-number a(X) is the dimension of the space of exact holomorphic differentials on X. We compute the a-number for infinite families of Fermat and Hurwitz curves. As our results apply to Hermitian curves, we recover a previous result by Gross [10].

Jacobian varieties of Hurwitz curves with automorphism group PSL(2,q)

The size of the automorphism group of a compact Riemann surface of genus g> 1 is bounded by 84.g 1/. Curves with automorphism group of size equal to this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL.2; q/. We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple.

Edmonds maps on Fricke-Macbeath curve

In 1985, L. D. James and G. A. Jones proved that the complete graph K n defines a clean dessin d’enfant (the bipartite graph is given by taking as the black vertices the vertices of K n and the white vertices as middle points of edges) if and only if n = p e , where p is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus g > 1 of these types of clean dessins d’enfant is g = 7 , obtained for p = 2 and e = 3 . In that case, there are exactly two such clean dessins d’enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7 ) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over Q , but both Edmonds maps cannot be defined over Q ; in fact they have as minimal field of definition the quadratic field Q (sqrt( − 7)) . It seems that no explicit models for the Edmonds maps over Q (sqrt( − 7)) are written in the literature. In this paper we start with an explicit model X for the Fricke-Macbeath curve provided by Macbeath, which is defined over Q ( e 2 π i / 7 ) , and we construct an explicit birational isomorphism L : X → Z , where Z is defined over Q (sqrt( − 7)) , so that both Edmonds maps are also defined over that field.

Polynomial bounds for Arakelov invariants of Belyi curves

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of the projective line with fixed branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's functions to deal with archimedean contributions. In the appendix, Peter Bruin proves an explicit version of Merkl's results.

A Canonical Curve of Genus 17

We compute equations for a Hurwitz curve of genus 17 and we conclude that the canonical ideal of any Hurwitz curve of genus 14 or 17 is generated by quadrics.

EXISTENCE OF COVERS WITH FIXED RAMIFICATION IN POSITIVE CHARACTERISTIC

We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at four points and compute the Hurwitz curve parametrizing these covers.

The quadrics through the Hurwitz curves of genus 14

A basis for the space of quadratic polynomials in the ideal of the canonical embedding of one of the three Hurwitz curves of genus 14 is found. Our method enables us to recover equations for other curves such as the Macbeath curve of genus 7 and the A-embedding of the Klein quartic in ℙ3.

Alexander modules of irreducible -groups

We give a complete description of the Alexander modules of knotted -manifolds in the sphere for and the Alexander modules of irreducible Hurwitz curves. This description is applied to investigate the properties of the first homology groups of cyclic coverings of the sphere and the complex projective plane branched respectively along knotted -manifolds and irreducible Hurwitz (in particular, algebraic) curves.

Alexander polynomials of Hurwitz curves

We study the properties of Alexander polynomials of Hurwitz curves and give a complete description of the set of the Alexander polynomials of irreducible Hurwitz curves in terms of their roots.

On Weierstrass points of Hurwitz curves

On Weierstrass points of Hurwitz curves

On symplectic coverings of the projective plane

We prove that a resolution of singularities of any finite covering of the projective complex plane branched along a Hurwitz curve , and possibly along the line “at infinity”, can be embedded as a symplectic submanifold in some projective algebraic manifold equipped with an integer Kähler symplectic form. (If has negative nodes, then the covering is assumed to be non-singular over them.) For cyclic coverings, we can realize these embeddings in a rational complex 3-fold. Properties of the Alexander polynomial of are investigated and applied to the calculation of the first Betti number , where is a resolution of singularities of an -sheeted cyclic covering of branched along , and possibly along the line “at infinity”. We prove that is even if is an irreducible Hurwitz curve but, in contrast to the algebraic case, may take any non-negative value in the case when consists of several components.

On the fundamental groups of the complements of Hurwitz curves

It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (or any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that there is a Hurwitz curve (resp. pseudoholomorphic curve) in such that the fundamental group of its complement is non-Hopfian and, therefore, this group is not residually finite. We also prove the existence of an irreducible non-singular algebraic curve and a bidisc such that the fundamental group is non-Hopfian.

Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves and (or, more generally, two curves with -type singularities) in the Hirzebruch surface with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if and are symplectic with respect to a compatible symplectic form.

A factorization formula for the full twist of double the number of strings

We give a formula for factorizing the full twist in the braid group in terms of four factorizations of the full twist in. This formula is used to construct a symplectic 4-manifold and two regularly homotopic generic coverings branched along cuspidal Hurwitz curves (without negative nodes) having different braid monodromy factorization types. The class of fundamental groups of complements of affine plane Hurwitz curves is described in terms of generators and defining relations.

On braid monodromy factorizations

We introduce and develop a language of semigroups over the braid groups to study the braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application, we give a new proof of Orevkov's theorem on the realization of bmf's over a disc by algebraic curves and show that the complexity of such a realization cannot be bounded in terms of the types of factors of the bmf. We also prove that the type of a bmf distinguishes Hurwitz curves with singularities of inseparable type up to -isotopy and -holomorphic cuspidal curves in up to symplectic isotopy.

Field of definition and Galois orbits for the Macbeath-Hurwitz curves

Hurwitz curves are Riemann surfaces with 84(g-1) automorphisms, g the genus. Defined over some number field they permit an obvious ${\rm Gal} (\overline {{\Bbb Q}}/{\Bbb Q})$ action. We investigate this action for the first known infinite series of Hurwitz curves, due to Macbeath, using the canonical model of the curves. As a result we obtain the minimal field of definition for these curves. The method can be extended to some other infinite series of modular curves for non-congruence subgroups.