论文标题
费马特类型曲线的p级
The p-rank of curves of Fermat type
论文作者
论文摘要
令$ \ mathbb {k} $为特征$ p> 0 $的代数关闭字段。代数曲线理论中的一个紧迫问题是确定（非单词，投影，不可约）曲线的$ p $ - 翼，$ \ mathcal {x} $ a $ \ mathbb {k} $，这个公寓不变的影响算术和$ \ nath的作用，以及$ \ mathcal的作用自动形态组$ \ operatatorName {aut}（\ Mathcal {x}）$在过去的几十年中已经注意到许多作者。在本文中，我们提供了Fermat类型$ y^m = x^n + 1 $ over $ \ mathbb {k} = \ bar {\ mathbb {f}} _ p $的$ p $ lank的广泛研究。我们在一般情况下确定了此不变的组合公式，并显示这是如何导致几种此类曲线的$ p $ lank的明确公式。通过插图，我们提出了二十多种曲线亚家族的明确公式，其中$ m $和$ n $通常以$ p $表示。我们还展示了如何使用该方法来计算其他类型的曲线的$ P $量。
Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$. A pressing problem in the theory of algebraic curves is the determination of the $p$-rank of a (nonsingular, projective, irreducible) curve $\mathcal{X}$ over $\mathbb{K}$, This birational invariant affects arithmetic and geometric properties of $\mathcal{X}$, and its fundamental role in the study of the automorphism group $\operatorname{Aut}(\mathcal{X})$ has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the $p$-rank of curves of Fermat type $y^m = x^n + 1$ over $\mathbb{K}=\bar{\mathbb{F}}_p$. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the $p$-rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where $m$ and $n$ are generally given in terms of $p$. We also show how the approach can be used to compute the $p$-rank of other types of curves.
