In this paper, we prove that a Thue equation F(x,y) = h, where h is an integer and F is a polynomial of degree n with integer coefficients and without repeated roots, has at most 2n^3 - 2n - 3 solutions provided that the Mordell-Weil rank of the Jacboian of the corresponding curve is less than (n-1)(n-2)/2. The proof uses the method of Coleman-Chabauty, extended here to apply to arbitrary regular models of curves, along with an explicit construction of a portion of a regular model for the curve corresponding to the Thue equation.
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.
Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mord…
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…
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 provably compute the full set of rational points on 1403 Picard curves defined over $\mathbb{Q}$ with Jacobians of Mordell-Weil rank $1$ using the Chabauty-Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes $p$-adic (Coleman) integrals on curves to points defined over number fields where the prime $p$ splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interes…
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.
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--…
We complete the computation of all $\mathbb{Q}$-rational points on all the $64$ maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty--Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels $N$, we classify all $\mathbb{Q}$-rational points as cusps, CM points (including their CM field and $j$-invariants) and exceptional ones. We further indicate how to use this …
Let $K$ be a number field with ring of integers $\mathcal O_{K}$. We prove that if $3$ does not divide $ [K:\mathbb Q]$ and $3$ splits completely in $K$, then the unit equation has no solutions in $K$. In other words, there are no $x, y \in \mathcal O_{K}^{\times}$ with $x + y = 1$. Our elementary $p$-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if $f \in \mathcal O_{K}[x]$ has a finite cyclic orbit in $\mathcal O_{K}$ of length…
