Related papers: On pointless diagonal Fermat curves

Let $p$ be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients \[3x^p + 8y^p + 21z^p =0\quad \text{ and } \quad 3x^p + 4y^p + 5z^p=0 \] do not admit non-trivial solutions for a set of exponents $p$…

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve $E/\Fp_q(C)$ over a function field over a finite field that have rank $\geq 2$, and for their average rank. The main tools are constructions and…

Let $E_{/\mathbb{Q}}$ be an elliptic curve with rank $E(\mathbb{Q})=0$. Fix an odd prime $p$, a positive integer $n$ and a finite abelian extension $K/\mathbb{Q}$ with rank $E(K) = 0$. In this paper, we show that there exist infinitely many…

Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results…

Let $\ell$ be an odd prime, $N \geq 1$ be an integer, and $\delta \geq 1$ be a $\ell^N$-th power free integer such that ${\rm ord}_{\ell}(\delta) = 0$ or $\ell \nmid {\rm ord}_{\ell}(\delta)$. In this paper, we give an explicit formula for…

We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the…

Within the scope of elementary number theory, we prove that, as the main result, if $1 \leq x < y < z$ are integers such that at least one of $y, z, x+y$ is prime then $x^{n}+y^{n} \neq z^{n}$ for every odd integer $n \geq 3$. This result…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the…

We study and explicitly construct some families of asymptotically exact sequences of algebraic function fields. It turns out that these families have an asymptotical class number widely greater than the general Lachaud - Martin-Deschamps…

For an imaginary quadratic field $k$ of class number $>1$, we prove that there are only finitely many isomorphism classes of rational indefinite quaternion division algebras $B$ such that the associated Shimura curve $M^B$ has $k$-rational…

Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower…

In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have…

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskr\k{e}cki--Stoll uses the…

Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of…

In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchm\'aros…

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…

Let $p$ and $q$ be distinct primes. Consider the Shimura curve $\mathcal{X}$ associated to the indefinite quaternion algebra of discriminant $pq$ over $\mathbb{Q}$. Let $J$ be the Jacobian variety of $\mathcal{X}$, which is an abelian…

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

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…