Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g-3. If K = Q, an explicit bound is 8 r g + 33 (g - 1) + 1. The proof is based on Chabauty's method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic `annulus' on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k). We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.Comment: 32 pages. v6: Some restructuring of the part of the argument relating to annuli in hyperelliptic curves (some section numbers have changed), various other improvements throughou

Simultaneous torsion in the Legendre family

We improve a result due to Masser and Zannier, who showed that the set $$\{\lambda \in {\mathbb C} \setminus \{0,1\} : (2,\sqrt{2(2-\lambda)}), (3,\sqrt{6(3-\lambda)}) \in (E_\lambda)_{\text{tors}}\}$$ is finite, where $E_\lambda \colon y^2 = x(x-1)(x-\lambda)$ is the Legendre family of elliptic curves. More generally, denote by $T(\alpha, \beta)$, for $\alpha, \beta \in {\mathbb C} \setminus \{0,1\}$, $\alpha \neq \beta$, the set of $\lambda \in {\mathbb C} \setminus \{0,1\}$ such that all points with $x$-coordinate $\alpha$ or $\beta$ are torsion on $E_\lambda$. By further results of Masser and Zannier, all these sets are finite. We present a fairly elementary argument showing that the set $T(2,3)$ in question is actually empty. More generally, we obtain an explicit description of the set of parameters $\lambda$ such that the points with $x$-coordinate $\alpha$ and $\beta$ are simultaneously torsion, in the case that $\alpha$ and $\beta$ are algebraic numbers that not 2-adically close. We also improve another result due to Masser and Zannier dealing with the case that ${\mathbb Q}(\alpha, \beta)$ has transcendence degree 1. In this case we show that $\#T(\alpha, \beta) \le 1$ and that we can decide whether the set is empty or not, if we know the irreducible polynomial relating $\alpha$ and $\beta$. This leads to a more precise description of $T(\alpha, \beta)$ also in the case when both $\alpha$ and $\beta$ are algebraic. We performed extensive computations that support several conjectures, for example that there should be only finitely many pairs $(\alpha, \beta)$ such that $\#T(\alpha, \beta) \ge 3$.Comment: 24 pages. v2: Improved 2-adic results, leading to more cases that can be treated explicitly. Used this to solve a problem considered in arXiv:1509.06573. Added some reference