In this paper, $\mathcal{X}$ is an algebraic curve of genus $\mathfrak{g}\ge 2$ defined over an algebraically closed field $\mathbb{K}$ of positive characteristic p, G is an automorphism group of $\mathcal{X}$ which fixes $\mathbb{K}$ element-wise, and, for a point $P\in \mathcal{X}$, $G_P$ is the subgroup of G which fixes P. The question “how large $G_P$ can be compared to $\mathfrak{g}$” has been the subject of several papers. We are concerned with the case where the second ramification group $G_P^{(2)}$ of $G_P$ is trivial. Under this condition Theorem 3.1 states that if $|G_P|>12(\mathfrak{g}-1)$ then $\mathcal{X}$ is either an ordinary hyperelliptic curve, or it has zero p-rank and $p\ne 3$. More precisely, up to birational equivalence, there exists a separable p-linearized polynomial $L(T)\in \mathbb{K}[T]$ of degree q such that an affine equation of $\mathcal{X}$ is $L(y)=ax+1/x$ with $a\in {\mathbb{K}}^{⁎}$ in the former case, and $L(y)=x^3+bx$ with $b\in \mathbb{K}$ in the latter case. In 1987 Nakajima proved that if $\mathcal{X}$ is an ordinary curve (more generally, the second ramification group of G is trivial for every $P\in \mathcal{X}$), then the order of G does not exceed $84\mathfrak{g}(\mathfrak{g}-1)$. We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from $84\mathfrak{g}(\mathfrak{g}-1)$ to $48{(\mathfrak{g}-1)}^2$.

在本文中， $\mathcal{X}$ 是属的代数曲线 $\mathfrak{G}\ge 2$ 在代数封闭域上定义 $\mathbb{ķ}$正特性的p，G ^是一个自同构组的$\mathcal{X}$ 修复 $\mathbb{ķ}$ 就元素而言 $P\in \mathcal{X}$， $G_P$是固定P的G的子组。问题“有多大$G_P$ 可以比较 $\mathfrak{G}$”已成为几篇论文的主题。我们关注第二分支小组的情况$G_P^{（2）}$ 的 $G_P$是微不足道的。在这种情况下，定理3.1指出：$|G_P|>12（\mathfrak{G}-\mathrm{1个}）$ 然后 $\mathcal{X}$是普通的超椭圆曲线，或者具有零p -rank和$p\ne 3$。更准确地说，直到双等式为止，存在一个可分离的p线性多项式$\mathrm{大号}（Ť）\in \mathbb{ķ}[Ť]$的度数为q的仿射方程$\mathcal{X}$ 是 $\mathrm{大号}（ÿ）=\mathrm{一种}X+\mathrm{1个}/X$ 与 $\mathrm{一种}\in {\mathbb{ķ}}^{⁎}$ 在前一种情况下，以及 $\mathrm{大号}（ÿ）=X^3+bX$ 与 $b\in \mathbb{ķ}$在后一种情况下。1987年，中岛证明了$\mathcal{X}$是一条普通曲线（更一般而言，G的第二个分支对于每个$P\in \mathcal{X}$），则G的阶数不超过$84\mathfrak{G}（\mathfrak{G}-\mathrm{1个}）$。我们证明了定理3.1和中岛计算的一些改进对中岛的边界有一点改进$84\mathfrak{G}（\mathfrak{G}-\mathrm{1个}）$ 至 $48{（\mathfrak{G}-\mathrm{1个}）}^2$。