Generalized Fermat, double Fermat and Newton sequences

In this paper, we discuss the relationship among the generalized Fermat, double Fermat, and Newton sequences. In particular, we show that every double Fermat sequence is a generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of double Fermat sequences, is closed under term-by-term multiplication. We also prove that every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that double Fermat sequences are Newton sequences generated by certain sequences of integers. An approach of symbolic dynamical systems is used to obtain congruence identities.     

Supersingular quotients of Fermat curves

We give formulas for the genera of all the possible quotient of a Fermat curve by a group of automorphisms in characteristic zero and for many classes of quotient curves also in positive characteristic. We use those results for giving evidence to the conjecture that, in any fixed positive characteristic, there should exist supersingular curves for any genus.     

Hasse-Witt matrices of Fermat curves

Let m be an integer with m3. Let K and K be perfect fields of characteristic p and p such that (p,m)=1 and (p,m)=1, respectively. Moreover let A and A be algebraic function fields over K and K defined by x^m+y^m=a(0, ak) and x^m+y^m=a(a0 ak), respectively. Put g=(m–1)(m–2)/2. Denote by M(K,p,a) and M(K,p,a) the Hasse-Witt matrices of A and A with respect to the canonical bases of holomorphic differentials. Then we show that if p+p0(mod.m) then rank M(K,p,a)+rank M(K,p,a)=g and if pp1 (mod.m) then rank M(K,p,a)=rank M(K,p,a).     

Periods of generalized Fermat curves

Let $k,n\ge 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface S that admits a subgroup of conformal automorphisms $H\le \text{Aut}(S)$ isomorphic to ${\mathbb{Z}}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\stackrel{?}{\mathbb{C}}$ 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.     

Hyperbolic uniformization of Fermat curves

The ground-breaking research on the uniformization of curves was conducted at the beginning of the last century. Nevertheless, there are few examples in the literature of algebraic curves for which an explicit uniformization is known. In this article we obtain an explicit uniformization of the Fermat curves F _N , for each . The results presented here are based in part on an earlier study of the second author [6] in which each Riemann surface F _N (ℂ) was described as a quotient of the complex disk by a Fuchsian group Γ. 2000 Mathematics Subject Classification Primary—11F03, 11F06; Secondary—11F30 This work was partially supported by MCYT BFM2000-0627 and BMF2003-01898.     

The arithmetic of Fermat curves

This research was supported in part by National Science Foundation grant DMS-9002095     

A characterization of maximal and minimal Fermat curves

In this article we provide a characterization of maximal and minimal Fermat curves using the classification of supersingular Fermat curves.     

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves F: aX ⁿ + bY ⁿ = Z ⁿ defined over F _q , we establish necessary and sufficient conditions for F to be F _q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F _q -Frobenius nonclassical cases, we determine explicit formulas for the number N _q (F) of F _q -rational points on F. For the remaining Fermat curves, nice upper bounds for N _q (F) are immediately given by the Stöhr–Voloch Theory.     

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 $S?{\mathbb{P}}^n$. 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.     

On generalizations of Fermat curves over finite fields and their automorphisms

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽_q of characteristic p>2. Assume that the 𝔽_q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups C_m and C_n of orders m and n prime to p, such that both quotient curves 𝒳∕C_n and 𝒳∕C_m are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.