Generalized Fermat Curves Research Articles

Smooth quotients of generalized Fermat curves

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $n,p \geq 2$ are integers such that $(p-1)(n-1)>2$, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms with quotient orbifold $S/H$ of genus zero with exactly $n+1$ cone points, each one of order $p$; in this case $H$ is called a generalized Fermat group of type $(p,n)$. In this case, it is known that $S$ is non-hyperelliptic and that $H$ is its unique generalized Fermat group of type $(p,n)$. Also, explicit equations for them, as a fiber product of classical Fermat curves of degree $p$, are known. For $p$ a prime integer, we describe those subgroups $K$ of $H$ acting freely on $S$, together with algebraic equations for $S/K$, and determine those $K$ such that $S/K$ is hyperelliptic.

Galois action on homology of generalized Fermat Curves

Abstract The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

Holomorphic differentials of generalized Fermat curves

A non-singular complete irreducible algebraic curve Fk,n, defined over an algebraically closed field K, is called a generalized Fermat curve of type (k,n), where n,k≥2 are integers and k is relatively prime to the characteristic p of K, if it admits a group H≅Zkn of automorphisms such that Fk,n/H is isomorphic to PK1 and it has exactly (n+1) cone points, each one of order k. By the Riemann-Hurwitz-Hasse formula, Fk,n has genus at least one if and only if (k−1)(n−1)>1. In such a situation, we construct a basis, called a standard basis, of its space H1,0(Fk,n) of regular forms, containing a subset of cardinality n+1 that provides an embedding of Fk,n into PKn whose image is the fiber product of (n−1) classical Fermat curves of degree k. For p=2, we obtain a lower bound (which is sharp for n=2,3) for the dimension of the space of exact one-forms, that is, the kernel of the Cartier operator. We also do this for (p,k,n)=(3,2,4).

Periods of generalized Fermat curves

Let k,n≥2 be integers. A generalized Fermat curve of type (k,n) is a compact Riemann surface S that admits a subgroup of conformal automorphisms H≤Aut(S) isomorphic to Zkn, such that the quotient surface S/H is biholomorphic to the Riemann sphere Cˆ and has n+1 branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.

On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Let G be the projective plane curve defined over Fq given byaXnYn−XnZn−YnZn+bZ2n=0, where ab∉{0,1}, and for each s∈{2,…,n−1}, let DsP1,P2 be the base-point-free linear series cut out on G by the linear system of all curves of degree s passing through the singular points P1=(1:0:0) and P2=(0:1:0) of G. The present work determines an upper bound for the number Nq(G) of Fq-rational points on the nonsingular model of G in cases where DsP1,P2 is Fq-Frobenius classical. As a consequence, when Fq is a prime field, the bound obtained for Nq(G) improves in several cases the known bounds for the number nP of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point P distinct from its vertices.

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let (S,H) be a generalized Fermat pair of the type (k,n). If F⊂S is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding S↪Pn. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.

Automorphisms of generalized Fermat curves

Let K be an algebraically closed field of characteristic p≥0. A generalized Fermat curve of type (k,n), where k,n≥2 are integers (for p≠0 we also assume that k is relatively prime to p), is a non-singular irreducible projective algebraic curve Fk,n defined over K admitting a group of automorphisms H≅Zkn so that Fk,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 Fk,n has genus gn,k>1 and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type (k,n) has a unique generalized Fermat group of type (k,n) if (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p).Generalized Fermat curves of type (k,n) can be described as a suitable fiber product of (n−1) classical Fermat curves of degree k. We prove that, for (k−1)(n−1)>2 (for p>0 we also assume that k−1 is not a power of p), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that p>0 and k−1 is a power of p, we use tools from the theory of complete projective intersections in order to prove that, for k and n+1 relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space.In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic p is either zero or p>kn−1.

Field of moduli of generalized Fermat curves of type [formula omitted] with an application to non-hyperelliptic dessins d'enfants

A generalized Fermat curve of type (k,3), where k≥2, is a closed Riemann surface admitting a group H≅Zk3 as a group of conformal automorphisms so that the quotient orbifold S/H is the Riemann sphere and it has exactly 4 cone points, each one of order k. Every genus one Riemann surface is a generalized Fermat curve of type (2,3) and, if k≥3, then a generalized Fermat curve of type (k,3) is non-hyperelliptic. For each generalized Fermat curve, we compute its field of moduli and note that it is a field of definition. Moreover, for k=e2iπ/p, where p≥5 is a prime integer, we produce explicit algebraic models over the corresponding field of moduli. As a byproduct, we observe that the absolute Galois group Gal(Q¯/Q) acts faithfully at the level of non-hyperelliptic dessins d'enfants. This last fact was already known for dessins of genus 0, 1 and for hyperelliptic ones, but it seems that the non-hyperelliptic situation is not explicitly given in the existent literature.

Automorphisms group of generalized Fermat curves of type [formula omitted

The determination of the full group of automorphisms of a closed Riemann surface is in general a very complicated task. For hyperelliptic curves, the uniqueness of the hyperelliptic involution permits one to compute these groups in a very simple manner. Similarly, as classical Fermat curves of degree k admit a unique subgroup of automorphisms isomorphic to Zk2, the determination of the group of automorphisms is not difficult.In this paper we consider a family of non-hyperelliptic Riemann surfaces, obtained as the fibre product of two classical Fermat curves of the same degree k, which exhibit behaviors of both elliptic and hyperelliptic curves. These curves, called generalized Fermat curves of type (k,3), are the highest regular abelian branched covers of orbifolds of genus zero with four cone points, all of the same order k. More precisely, a generalized Fermat curve of type (k,3) is a closed Riemann surface S admitting a group H<Aut(S), called a generalized Fermat group of type (k,3), so that H≅Zk3 and S/H is an orbifold with signature (0,4;k,k,k,k). In this paper we prove the uniqueness of generalized Fermat groups of type (k,3). In particular, this allows the explicit computation of the full group of automorphisms of S.

ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.

FIELD OF MODULI OF GENERALIZED FERMAT CURVES

As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and only if $S$ admits an anticonformal automorphism. In the case that the field of moduli of $S$ is ${\mathbb R}$, then $S$ can be defined over the field of moduli if and only if $S$ admits an anticonformal involution. It may happen that the field of moduli is not a field of definition. In this paper, we consider certain class of closed Riemann surfaces, called generalized Fermat curves. These surfaces are the highest Abelian branched cover of certain orbifolds. In this class of Riemann surfaces, we study the problem of deciding when the field of moduli is ${\mathbb R}$ and when, in such a case, it is a field of definition.

Generalized Fermat curves

A closed Riemann surface S is a generalized Fermat curve of type ( k , n ) if it admits a group of automorphisms H ≅ Z k n such that the quotient O = S / H is an orbifold with signature ( 0 , n + 1 ; k , … , k ) , that is, the Riemann sphere with ( n + 1 ) conical points, all of same order k. The group H is called a generalized Fermat group of type ( k , n ) and the pair ( S , H ) is called a generalized Fermat pair of type ( k , n ) . We study some of the properties of generalized Fermat curves and, in particular, we provide simple algebraic curve realization of a generalized Fermat pair ( S , H ) in a lower-dimensional projective space than the usual canonical curve of S so that the normalizer of H in Aut ( S ) is still linear. We (partially) study the problem of the uniqueness of a generalized Fermat group on a fixed Riemann surface. It is noted that the moduli space of generalized Fermat curves of type ( p , n ) , where p is a prime, is isomorphic to the moduli space of orbifolds of signature ( 0 , n + 1 ; p , … , p ) . Some applications are: (i) an example of a pencil consisting of only non-hyperelliptic Riemann surfaces of genus g k = 1 + k 3 − 2 k 2 , for every integer k ⩾ 3 , admitting exactly three singular fibers, (ii) an injective holomorphic map ψ : C − { 0 , 1 } → M g , where M g is the moduli space of genus g ⩾ 2 (for infinitely many values of g), and (iii) a description of all complex surfaces isogenous to a product with invariants p g = q = 0 and covering group equal to Z 5 2 or Z 2 4 .

Some comments on the solvable chiral potts model

We present a simple proof of the conjecture produced by Baxter, Perk and Au-Yang on the structure of the normalization factorR(p, q, r) corresponding to their new solution of the star-triangle equation related with the generalized Fermat curve. Some important properties of the underlying curvex N y N+x N+y N+1/k 2=0 for theN=3 state case are also established. Particularly, we calculate exactly its matrix of theb-periods for some normalized basis of holomorphic differentials. We also show that associated four-dimensional theta function may be decomposed into a sum containing 12 terms, each term being the product of four one-dimensional theta functions. We also derive Picard-Fuchs equations for the periods of holomorphic differentials of the same curve. The remarkable appearance of the hypergeometric functions in our answers seems to be closely related with an expression for the groundstate energy per site, obtained for the superintegrable case by Albertini, Perk, and McCoy and independently by Baxter, although for a moment the connection is not clear.