The Weil group of a hyper-class formation

Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves

In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.

Hypersurfaces and the Weil conjectures

We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the l-adic Fourier transform.

曲线上 $\mathbf{P}^n$-丛中超曲面的 Chevalley-Weil 公式与函数域塔中的 Mordell-Weil 秩

Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers

本文研究曲线上射影丛中超曲面在 Galois 基变换下的 Hodge 上同调群结构，给出 Chevalley-Weil 型公式，并借此几何证明椭圆曲面 Mordell-Weil 秩的 Pal 上界，同时比较其与 Shioda-Tate 公式界的强弱关系。

关于特征为2的有限域上超椭圆曲线的Weil多项式

On Weil Polynomials of Hyperelliptic Curves over Finite Fields of Characteristic 2

本文通过模2剩余类约束给出特征2有限域上超椭圆雅可比矩阵存在性的新阻碍条件，证明Weil系数奇偶性渐近等分布，并据此排除大量3维和4维同源类，同时提出运行时间显著改进的任意亏格超椭圆曲线枚举算法。

Splitting curves on a rational ruled surface, the Mordell-Weil groups of hyperelliptic fibrations and Zariski pairs

Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…

Hasse-Weil Zeta 函数模素数

Hasse-Weil Zeta Functions Modulo a Prime

本文通过有限域上分离有限型概形的有限态射，确定了在素数阶伽罗瓦群作用下zeta函数的模r约化，并应用于曲线及超椭圆/超椭圆曲线的分子。