On the Jacobian variety of the Fermat curve

Revista Colombiana De Matematicas, 2013

A generalized Fermat curve of type (p, n) is a closed Riemann surface S admitting a group H ∼ = Z n p of conformal automorphisms with S/H being the Riemann sphere with exactly n + 1 cone points, each one of order p. If (p − 1)(n − 1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by AutH (S) the normalizer of H in Aut(S). If p is a prime, and either (i) n = 4 or (ii) n is even and AutH (S)/H is not a non-trivial cyclic group or (iii) n is odd and AutH (S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3, 4}, then we also compute the field of moduli of S.

Journal of Algebra, 1986

Let X;(p) denote the Fermat variety of dimension r and of degree m detined over a finite field F,, with q= p' elements. For a fixed m, we prove that certain arithmetical invariants of XL(p), such as the stable rank of X;,,(p), and the Picard number of P,(p), depend only on the congruence class of p modulo m. The results are extended also to products of Fermat varieties over IF,, and also to finite quotients of a Fermat variety over IF,,. @Z

Revista Colombiana De Matematicas, 2013

A generalized Fermat curve of type (p, n) is a closed Riemann surface S admitting a group H ∼ = Z n p of conformal automorphisms with S/H being the Riemann sphere with exactly n + 1 cone points, each one of order p. If (p − 1)(n − 1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by AutH (S) the normalizer of H in Aut(S). If p is a prime, and either (i) n = 4 or (ii) n is even and AutH (S)/H is not a non-trivial cyclic group or (iii) n is odd and AutH (S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3, 4}, then we also compute the field of moduli of S.

Canadian Mathematical Bulletin, 2009

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

1999

Let GF(q) be the Galois field of order q"pF, and let m53 be an integer. An explicit formula for the number of GF(qK)-rational points of the Fermat curve XL#½L#ZL"0 is given when n divides (qK!1)/(q!1) and p is sufficiently large with respect to (qK!1)/(n(q!1)).

arXiv: General Mathematics, 2014

This paper develops a framework of algebra whereby every Diophantine equation is made quickly accessible by a study of the corresponding row entries in an array of numbers which we call the Newtonian triangles. We then apply this framework to the understandimg of the Fermat's Last Theorem and discuss some of its direct consequences.

International Journal of Mathematics, 2016

We obtain the trace map images of the values of certain harmonic volumes for some cyclic quotients of [Formula: see text]th Fermat curves. These provide the algorithm showing that the algebraic cycles, called [Formula: see text]th Ceresa cycles, are not algebraically equivalent to zero in their Jacobian varieties. We apply the method to curves for prime [Formula: see text] with [Formula: see text] modulo [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text].

Bulletin of the Polish Academy of Sciences Mathematics 01/2008; 56:199-206.

We study the family of curves Fm (p) : x^p + y^p = m, where p is an odd prime and m is a pth power free integer. We prove some results about distribution of root numbers of the L-functions of the hyperelliptic curves associated to the Fm (p). As a corollary we obtain that the jacobians of the curves Fm (5) with even analytic rank and those with odd analytic rank are equally distributed.

2021

Let p be a prime, let r and q be powers of p, and let a and b be relatively prime integers not divisible by p. Let C/Fr(t) be the superelliptic curve with affine equation y b + x = t − t. Let J be the Jacobian of C. By work of Pries–Ulmer [PU16], J satisfies the Birch and SwinnertonDyer conjecture (BSD). Generalizing work of Griffon–Ulmer [GU20], we compute the L-function of J in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of J appearing in BSD, including the rank of the Mordell–Weil group J(Fr(t)), the Faltings height of J , and the Tamagawa numbers of J in terms of the parameters a, b, q. For any p and r, we show that for certain a and b depending only on p and r, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as q varies through powers of p. Under a different set of criteria on a and b, we prove that the order of the Tate–Shafarevich group X(J) g...

The automorphism group of the generalized Fermat $F_{k,n}$ curves is studied. We use tools from the theory of complete projective intersections in order to prove that every automorphism of the curve can be extended to an automorphism of the ambient projective space. In particular if $k-1$ is not a power of the characteristic, then a conjecture of of Y. Fuertes, G. Gonz\'alez-Diez, R. Hidalgo, M. Leyton is proved.