We give a generalization of the method of "Elliptic Curve Chabauty" to higher genus curves and their Jacobians. This method can sometimes be used in conjunction with covering techniques and a modified version of the Mordell-Weil sieve to provide a complete solution to the problem of determining the set of rational points of an algebraic curve $Y$.
We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman t…
We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.
Let $X/\mathbb{Q}$ be a curve of genus $g \ge 2$ with Jacobian $J$ and let $\ell$ be a prime of good reduction. Using Selmer varieties, Kim defines a decreasing sequence \[ X(\mathbb{Q}_\ell) \supseteq X(\mathbb{Q}_\ell)_1 \supseteq X(\mathbb{Q}_\ell)_2 \supseteq \cdots \] all containing the rational points of $X$. Thanks to the work of Coleman, the `Chabauty set' $X(\mathbb{Q}_\ell)_1$ is known to be finite provided the Mordell--Weil rank of $J$ is smaller than $g$. In this case one has a practical strategy that often succeeds in computing the set of rational points of $X$. Balakrishnan and D…
We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek's extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.
Building on work of Balakrishnan, Dogra, and of the first author, we provide some improvements to the explicit quadratic Chabauty method to compute rational points on genus $2$ bielliptic curves over $\mathbb{Q}$, whose Jacobians have Mordell-Weil rank equal to $2$. We complement this with a precision analysis to guarantee correct outputs. Together with the Mordell-Weil sieve, this bielliptic quadratic Chabauty method is then the main tool that we use to compute the rational points on the $411$ locally solvable curves from the LMFDB which satisfy the aforementioned conditions.
The main point of the paper is to take the explicit motivic Chabauty-Kim method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author and make it work for non-rational curves. In particular, we calculate the abstract form of an element of the Chabauty-Kim ideal for $\mathbb{Z}[1/\ell]$-points on a punctured elliptic curve, and lay some groundwork for certain kinds of higher genus curves. For this purpose, we develop an "explicit Tannakian Chabauty-Kim method" using $\mathbb{Q}_{p}$-Tannakian categories of Galois representations in place of $\mathbb{Q}$-linear motives. In future…
We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the first explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell-Weil rank larger than its genus.
Faltings' theorem states that curves of genus $g \geq 2$ have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than $g$, can be used to give a good effective bound on the number of rational points of curves of genus $g > 1$. We draw ideas from nonarchimedean geometry to show that we can also give an effect…
The Chabauty--Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell--Weil rank less than their genus. We show how the Chabauty--Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a nonabelian generalisation of Coleman's effective Chabauty theorem.
We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell--Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or nontrivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N…
In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is prove…
We compute rational points on genus $3$ odd degree hyperelliptic curves $C$ over $\mathbb{Q}$ that have Jacobians of Mordell-Weil rank $0$. The computation applies the Chabauty-Coleman method to find the zero set of a certain system of $p$-adic integrals, which is known to be finite and include the set of rational points $C(\mathbb{Q})$. We implemented an algorithm in Sage to carry out the Chabauty-Coleman method on a database of $5870$ curves.
Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound $\# X(K)$ when $\mathrm{rank}\ J(K) < g$. We study limitations of a variant called `Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty typically bounds $\# X(K)$ when $\mathrm{rank}\ J(K) \leq [K:\mathbb{Q}] (g - 1)$, but fails in the presence of a subgroup obstruction, a high-rank subgroup scheme of $\mathrm{Res}_{K/\mathbb{Q}} J$ which intersects the image of $\mathrm{Res}_{K/\mathbb{Q}} X$ in higher-than-expected dimen…
We study the Selmer varieties of smooth projective curves of genus at least two defined over $\mathbb{Q}$ which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov-Tschinkel and Poonen on unramified correspondences, we deduce that any cover of $\mathbf{P}^1$ with solvable Galois group, and in particular any superelliptic curve over $\mathbb{Q}$, has only finitely many rational points over $\mathbb{Q}$.
Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve $C$ of genus $g\ge 2$ defined over a number field $F$, with Jacobian of rank at most $g-1$. Namely, in the case $F=\mathbb{Q}$, if $p>2g$ is a prime of good reduction, then the number of rational points of $C$ is at most the number of $\mathbb{F}_p$-points plus a contribution coming from the canonical class of $C$. We prove a result analogous to Coleman's bound in the case of a hyperbolic surface $X$ over a number field, embedded in an abelian variety $A$ of rank at mos…
We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus $g \ge 2$ over the rationals whose Jacobian has Mordell-Weil rank $g$ and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve $X_{s}(13)$, completing the classification of non-CM elliptic curves over $\mathbf{Q}$ with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.
We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--…
Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus $2$ quintic curves over $\mathbb{Q}$ of Mordell-Weil rank $2$ or $3$ whos…
Given a quintic number field $K/\mathbb{Q}$, we study the set of irreducible trinomials, polynomials of the form $x^{5} + ax + b$, that have a root in $K$. We show that there is a genus four curve $C_{K}$ whose rational points are in bijection with such trinomials. This curve $C_{K}$ maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on $C_{K}$ using elliptic curve Chabauty.
Let $C$ be a genus $2$ hyperelliptic curve over a number field $K$, with a Weierstrass point $\infty$ at infinity, let $J$ be its Jacobian, let $Θ$ be the theta divisor with respect to $\infty$, and let $p$ be any prime number. We give an explicit construction of a $p$-adic height $h_p\colon J(\overline{\mathbb{Q}})\to \mathbb{Q}_p$ by means of $p$-adic analogues of Néron functions of divisor $2Θ$. We define such Néron functions using division polynomials and a generalisation of Blakestad's $p$-adic sigma function on the formal group of $J$. We prove that our $p$-adic Néron function $λ_v$ at a…
Bruin and Najman, Ozman and Siksek, and Box described all the quadratic points on the modular curves of genus $2\leq g(X_0(n)) \leq 5$. Since all the hyperelliptic curves $X_0(n)$ are of genus $\leq 5$ and as a curve can have infinitely many quadratic points only if it is either of genus $\leq 1$, hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves $X_0(n)$ naturally arises; this question has recently also been posed by Mazur. We answer Mazur's question completely and describe the quadratic points on all the bielliptic modular curves $X…
Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this …
We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d < g$ is even, we conditionally bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and B_3=114$. Our proofs proceed by refining recent work of Park, which applied tropical geometry to symmetric power Chabauty, and then applying resul…
Let $C$ be a curve of genus at least three defined over a number field, and let $r$ be the rank of the rational points of its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on $C$ by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on $r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results…
We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs $(N, r)$ such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an …
We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the rational points on $C_{k}$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k \in \mathbb{Q}$.
Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel--Jacobi map. For fixed $X$ this quantity was conjectured to be finite by Manin--Mumfor…
