Let C be a curve defined over a number field K and write g for the genus of C and J for the Jacobian of C. Let $n \ge 2$. We say that an algebraic point $P \in C(\overline {K})$ has degree n if the extension $K(P)/K$ has degree n. By the Galois group of P we mean the Galois group of the Galois closure of $K(P)/K$ which we identify as a transitive subgroup of $S_n$. We say that P is primitive if its Galois group is primitive as a subgroup of $S_n$. We prove the following ‘single source’ theorem for primitive points. Suppose $g>(n-1)^2$ if $n \ge 3$ and $g \ge 3$ if $n=2$. Suppose that either J is simple or that $J(K)$ is finite. Suppose C has infinitely many primitive degree n points. Then there is a degree n morphism $\varphi : C \rightarrow \mathbb {P}^1$ such that all but finitely many primitive degree n points correspond to fibres $\varphi ^{-1}(\alpha )$ with $\alpha \in \mathbb {P}^1(K)$.

We prove, moreover, under the same hypotheses, that if C has infinitely many degree n points with Galois group $S_n$ or $A_n$, then C has only finitely many degree n points of any other primitive Galois group.

We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$-covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.

In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels–Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where J is the Jacobian variety of a genus two curve under the assumption that all points in J[2] are K-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating that we can improve the rank bound given by a 2-descent via computing the Cassels–Tate pairing.

For N integer $\ge 1$, K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth Fermat curve. We show that the Manin–Drinfeld principle holds for $N=3$, but not for $N=5$. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$, for n even integer, both dominate $X(2)$, which produces a morphism between Jacobians $J_N\rightarrow J(n)$. We prove that the latter has image $0$ or an elliptic curve of j-invariant $0$. In passing, we give a description of the homology of $X^{\prime }_{N}$.

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.

Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$).

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

Let K be an algebraic number field. We investigate the K-rational distance problem and prove that there are infinitely many nonisomorphic cubic number fields and a number field of degree n for every $n\geq 2$ in which there is a point in the plane of a unit square at K-rational distances from the four vertices of the square.

Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$, almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$. We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$. We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$. On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$ -function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.

We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

We describe the construction of a database of genus- $2$ curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated $L$ -function. This data has been incorporated into the $L$ -Functions and Modular Forms Database (LMFDB).

Given a sextic CM field $K$ , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$ , and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field  $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo  $p$ .

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

We give a separability criterion for the polynomials of the form

$$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$

$$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$

Let $Q(N;q,a)$ be the number of squares in the arithmetic progression $qn+a$ , for $n=0$ , $1,\ldots,N-1$ , and let $Q(N)$ be the maximum of $Q(N;q,a)$ over all non-trivial arithmetic progressions $qn + a$ . Rudin’s conjecture claims that $Q(N)=O(\sqrt{N})$ , and in its stronger form that $Q(N)=Q(N;24,1)$ if $N\ge 6$ . We prove the conjecture above for $6\le N\le 52$ . We even prove that the arithmetic progression $24n+1$ is the only one, up to equivalence, that contains $Q(N)$ squares for the values of $N$ such that $Q(N)$ increases, for $7\le N\le 52$ ( $N=8,13,16,23,27,36,41$ and $52$ ).

Supplementary materials are available with this article.

It is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y0(n). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X1+X2=1 or the hyperbola X1X2=1 it may be shown quite quickly that there are no imaginary quadratic τ1,τ2 with j(τ1)+j(τ2)=1 or j(τ1)j(τ2)=1, where j is the classical modular function.

We study the arithmetic of abelian varieties over where is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over to homomorphisms of other Jacobians over . Our methods also yield completely explicit points on elliptic curves with unbounded rank over and a new construction of elliptic curves with moderately high rank over .