Let $F$ be a separable integral binary form of odd degree $N \geq 5$. A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree-$N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family $\mathscr {F}_N(f_0)$, more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$, respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g\geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve, or a formal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y, and the exact sequence of $\pi _1(X)$ splits, then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite.

The problem of classifying elliptic curves over $\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$. In this article, we classify genus $2$ curves C defined over ${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$-parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.

In a previous paper, we discussed Frobenius-projective structures on projective smooth curves in positive characteristic and established a relationship between pseudo-coordinates and Frobenius-indigenous structures by means of Frobenius-projective structures. In the present paper, we discuss an “affine version” of this study of Frobenius-projective structures. More specifically, we discuss Frobenius-affine structures and establish a similar relationship between Tango functions and Frobenius-affine-indigenous structures by means of Frobenius-affine structures. Moreover, we also consider a relationship between these objects and Tango curves.

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over $\mathbb{Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$.

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.

In this paper, we study bounds for the number of rational points on twists $C'$ of a fixed curve $C$ over a number field ${\mathcal K}$, under the condition that the group of ${\mathcal K}$-rational points on the Jacobian $J'$ of $C'$ has rank smaller than the genus of $C'$. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form $2r + c$, where $r$ is the rank of $J'({\mathcal K})$ and $c$ is a constant depending on $C$. For the proof, we use a refinement of the method of Chabauty–Coleman: the main new ingredient is to use it for an extension field of ${\mathcal K}_v$, where $v$ is a place of bad reduction for $C'$.

We show that isomorphisms between arithmetic fundamental groups of hyperbolic curves over p-adic local fields preserve the decomposition groups of all closed points (respectively, closed points arising from torsion points of the underlying elliptic curve), whenever the hyperbolic curves in question are isogenous to hyperbolic curves of genus zero defined over a number field (respectively, are once-punctured elliptic curves [which are not necessarily defined over a number field]). We also show that, under certain conditions, such isomorphisms preserve certain canonical “integral structures” at the cusps [i.e., points at infinity] of the hyperbolic curve.

We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$, when $g$ is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension.

We prove that if q is a power of an odd prime, then there is no genus-2 curve over $\mathbf{F}_q$ whose Jacobian has characteristic polynomial of Frobenius equal to $x^4 + (2 - 2q)x^2 + q^2$. Our proof uses the Brauer relations in a biquadratic extension of $\mathbb{Q}$ to show that every principally polarized abelian surface over $\mathbf{F}_q$ with the given characteristic polynomial splits over $\mathbf{F}_{q^2}$ as a product of polarized elliptic curves.

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Soient $r$ et $p$ deux nombres premiers distincts, soit $K\,=\,\mathbb{Q}(\cos \,\frac{2\pi }{r})$ , et soit $\mathbb{F}$ le corps résiduel de $K$ en une place au-dessus de $p$. Lorsque l’image de $(2\,-\,2\,\cos \,\frac{2\pi }{r})$ dans $\mathbb{F}$ n’est pas un carré, nous donnons une construction géométrique d’une extension réguliere de $K\left( t \right)$ de groupe de Galois $\text{PS}{{\text{L}}_{2}}(\mathbb{F})$ . Cette extension correspond à un revêtement de ${{\mathbb{P}}^{1}}/k$ de « signature $\left( r,\,p,\,p \right)$ » au sens de [3, sec. 6.3], et son existence est prédite par le critère de rigidité de Belyi, Fried, Thompson et Matzat. Sa construction s’obtient en tordant la representation galoisienne associée aux points d’ordre $p$ d’une famille de variétés abéliennes à multiplications réelles par $K$ découverte par Tautz, Top et Verberkmoes [6]. Ces variétés abéliennes sont définies sur un corps quadratique, et sont isogènes à leur conjugué galoisien. Notre construction généralise une méthode de Shih [4], [5], que l’on retrouve quand $r\,=\,2$ et $r\,=\,3$.