Deterministically generating Picard groups of hyperelliptic curves over finite fields

From Picard groups of hyperelliptic curves to class groups of quadratic fields

Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree $0$ on $C$ which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas.

K3曲面上的超椭圆曲线

Hyperelliptic curves on $K3$ surfaces

该研究证明皮卡秩为1的K3曲面上任意多个一般点可由超椭圆曲线连接，得到小对角的Beauville–Voisin分解及挠阶新界，还将Corvaja–Zucconi定理推广到任意次数K3曲面，证明其二次点在有限扩张后稠密。

通过C-对称矩阵研究超椭圆曲线皮卡群的4秩分布

4-rank distribution of Picard groups of hyperelliptic curves via $C$-symmetric matrices

本文通过C-对称矩阵研究超椭圆曲线皮卡群的4秩分布，揭示了有限域上超椭圆曲线的皮卡群4秩的极限分布特性。

Ideal class groups and torsion in Picard groups of varieties

We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\mathbb{Q}$ to obtain class groups for number fields $\mathbb{Q}(P)$, $P\in V(\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its…

An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over $\mathbb{C}$. More precisely, we elaborate on an algorithm that, given a small period matrix $Ω\in \mathbb{C}^{3\times 3}$ corresponding to a principally polarized abelian threefold equipped with an automorphism of order $3$, returns a Legendre-Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike-Weng in [Math. Comp., 74(249):499-518, 2005] which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain (numericall…

Kummer varieties and their Brauer groups

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with larg…