In this paper, we consider the family of hyperelliptic curves over $\mathbb{Q}$ having a fixed genus $n$ and a marked rational non-Weierstrass point. We show that when $n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as $n$ tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of $5/2$ on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $\mathbb{Q}$ with good reduction away from 3, up to $\mathbb{Q}$-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.

We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over  $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

Let $C\subset C_1\times C_2$ be a curve of type $(d_1,d_2)$ in the product of the two curves $C_1$ and $C_2$. Let $\nu$ be a positive integer. We prove that if a certain inequality involving $d_1,\ d_2,\ \nu$, and the genera of the curves $C_1,\ C_2$, and $C$ is satisfied, then the set of points $\{P\in C(\bar{k})\mid[k(P):k]\leq \nu\}$ is finite for any number field $k$. We prove a similar result for integral points of bounded degree on $C$. These results are obtained as consequences of an inequality of Vojta which generalizes the Roth–Wirsing theorem to curves.

It is shown that a $\scriptstyle \overline {\mathbb Q}$-curve of genus $g$ and with stable reduction (in some generalized sense) at every finite place outside a finite set $S$ can be defined over a finite extension $L$ of its field of moduli $K$ depending only on $g$, $S$ and $K$. Furthermore, there exist $L$-models that inherit all places of good and stable reduction of the original curve (except possibly for finitely many exceptional places depending on $g$, $K$ and $S$). This descent result yields this moduli form of the Shafarevich conjecture: given $g$, $K$ and $S$ as above, only finitely many $K$-points on the moduli space ${\cal M}_g$ correspond to $\scriptstyle \overline {\mathbb Q}$-curves of genus $g$ and with good reduction outside $S$. Other applications to arithmetic geometry, like a modular generalization of the Mordell conjecture, are given.