In this paper we show a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field $k$. The method is based on a correspondence between twists and solutions to a Galois embedding problem. When in addition, this curve is non-hyperelliptic we show how to compute equations for the twists. If $k = \mathbb{F}_{q}$ the method then becomes an algorithm, since in this case, the Galois embedding problems that appear are known how to be solved. As an example we compute the set of twists of the non-hyperelliptic genus $6$ curve $x^{7}-y^{3}z^{4}-z^{7} = 0$ when we consider it defined over a number field such that $[k(ζ_{21}):k] = 12$. For each twist equations are exhibited.
We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus.
We study a class of semistable ordinary hyperelliptic curves over 2-adic fields and the special fibre of their minimal regular model. We show that these curves can be controlled using `cluster pictures', similarly to the case of odd residue characteristic.
Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves $X_0(D,N)$. In this paper we compute the $\mathbb{Q}$-rational points on the Atkin-Lehner quotients of these curves using a variety of techniques. We also determine which rational points are CM for many of these curves.
2-Selmer groups of even hyperelliptic curves over function fields
本文计算了函数域上偶超椭圆曲线族的2-Selmer群的平均长。结果将通过基于群$G=\text{PSO}(2n+2)$的Vinberg表示和希钦纤维化的几何方法获得。与Arul Shankar和Xiaoheng Wang [超椭圆曲线上有显著非Weierstrass点的有理点，Compos.数学、154（1）：188-222，2018]，我们提供了平均值的上界和下界。然而，如果我们限制到族的横向超椭圆曲线，我们得到精确的数字6。
We give a condition for a hyperelliptic curve $C$ over a local field $K$ to be locally soluble, on the condition that $C$ obtains semistable reduction after a tame extension of $K$, and that the residue field $k$ is sufficiently large relative to the genus of the curve. The condition is presented in terms of the cluster picture of $C$, a combinatorial object which determines much of the local arithmetic of $C$.
Let $K$ be a complete discretely valued field of residue characteristic not $2$ and $O_K$ its ring of integers. We explicitly construct a regular model over $O_K$ with strict normal crossings of any hyperelliptic curve $C/K:y^2=f(x)$. For this purpose, we introduce the new notion of ''MacLane cluster picture'', that aims to be a link between clusters and MacLane valuations.
Algorithmic study of superspecial hyperelliptic curves over finite fields
本文提出了研究超特殊超椭圆曲线的算法方法。本文提出的算法是：一个计算有限域上亏格为$\mathbb{F}_q$的超特殊超椭圆曲线的算法，以及一个计算有限域上（不一定是超特殊）超椭圆曲线的自同构群的算法。第一种算法适用于任何$(g,q)$，使得$q$和$2g+2$互质且$q>2g+1$。作为应用，我们在计算机代数系统Magma上实现了亏格为$g=4$的超特殊超椭圆曲线在$\mathbb{F}_{p}$上对$11 \leq p \leq 23$和在$\mathbb{F}_{p^2}$上对$11 \leq p \leq 19$的计算.此外，我们发现了最大的超椭圆曲线和最小的超椭圆曲线在$\mathbb{F}_{p^2}$从列举的超特殊。第二个算法计算一个自同构作为一个具体的元素（的商）的线性组在一般线性组的程度$2$。
Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision. Consequently, after having possibly lifted the problem in the $p$-adics, we derive fast algorithms for computing explicitly Cantor's division polynomials of hyperelliptic curves defined over finite fields.
We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
