Improved rank bounds from 2-descent on hyperelliptic Jacobians

We describe a qualitative improvement to the algorithms for performing [Formula: see text]-descents to obtain information regarding the Mordell–Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity.

Constructing genus-3 hyperelliptic Jacobians with CM

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 $p$.

Rational points on certain families of symmetric equations

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves x2m + axm + aym + y2m = b whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe Xd(ℚ) for certain d ≥ 3, where Xd : Td(x) + Td(y) = 1 and Td is the monic Chebychev polynomial of degree d. Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus 3 curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of x4 - 4x2 - 4y2 + y4 = -6.