相关论文: Statistics of Different Reduction Types of Fermat …

The aim of this paper is to classify reduction types of algebraic curves. Reduction types capture the discrete invariants of fibres in one-dimensional families of curves, and they have been described in genus 1, 2 and 3. For fixed genus…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$. A pressing problem in the theory of algebraic curves is the determination of the $p$-rank of a (nonsingular, projective, irreducible) curve $\mathcal{X}$ over…

Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=\delta$ where $\delta$ is an…

Let $p\geq 3$ be a prime number. A Fermat curve over $\mathbb{Q}$ of exponent $p$ is defined by an equation of the shape $ax^p+by^p+cz^p=0$, where $a,b,c$ are non-zero rational numbers. We prove in this article that there exist infinitely…

We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational…

We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $\p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of…

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of…

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z)…

We solve the Diophantine equations $x^5 + y^5 = dz^p$ with $d=2, 3$ for a set of prime numbers of density 1/4, 1/2, respectively. The method consists in relating a possible solution to another Diophantine equation and solving the later by…

In this article we improve the upper bound for the arithmetic self-intersection number of the dualizing sheaf of the minimal regular model for the Fermat curves $F_p$ of prime exponent.

Despite their simple definition as $\mathfrak{q}_p(b):=\frac{b^{p-1}-1}{p} \pmod p$, for $0\le b \le p^2-1$ and $\gcd(b,p)=1$, and their regular arrangement in a $p\times(p-1)$ Fermat quotient matrix $\mathtt{FQM}(p)$ of integers from…

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are…

We show that the existence of a non-trivial solution of $x^n+y^n=p^n$, with $p$ a prime number, is equivalent to the existence of a solution of a certain (over-determined) system of $(n-1)$-recursion relations ("zipper" equations) in…

This paper establishes a constructive link between the first slope of Artin-Schreier curves X_f: y^p-y=f(x) and the p-adic weight of the support of f(x). If the maximal p-adic weight element v in Supp(f) is unique, we show that the first…

Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of…

Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ${\bf Q}$ given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there…

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…

A somewhat pretentious presentation of number systems (N, Z, Q, R, C, Q_p, >...). The problem of a p-adic characterisation of good-reduction p-adic curves is posed.

We follow the ideas of Darmon's program for solving infinite families of generalised Fermat equations of signatures $(p,p,r)$ and $(r,r,p)$, where, $r$ is a fixed prime and $p$ is varying. We do so by introducing a common framework for both…

We classify all geometric torsion points on the Fermat quotients $y^n = x^d + 1$ where $n, d \ge 2$ are coprime. In addition, we classify all geometric torsion points on the generic superelliptic curve $y^n = (x - a_1) \cdots (x - a_d)$,…