AI 中文总结
本文在特征2中构造亏格q+1的普通双椭圆曲线，其自同构群达到极值4q，实现Giulietti-Korchmáros分类中的情形(ib)，并给出该情形的完全刻画与参数化。
AI 中文摘要
设 $k$ 为特征 $2$ 的代数闭域，$q=2^h$，$h\ge3$。我们构造亏格为 $q+1$ 的普通双椭圆曲线 $X$，使得 \\[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \\] 这些曲线实现了 Giulietti--Korchmáros 分类中的情形 \textup{(ib)}，并给出了回答 Korchmáros 所提问题的一个无限族。更一般地，我们证明情形 \textup{(ib)} 中的每条曲线都来自同一构造。情形 \textup{(ib)} 恰好出现在亏格 $g=2^h+1$（$h\ge2$）中。对于 $g\ge9$，我们确定完全自同构群；在亏格 $5$ 时，我们确定其 Sylow $2$-子群，但不声称完全自同构群。对于每个固定的 $q\ge8$，亏格 $q+1$ 的情形 \textup{(ib)} 中的同构类由 $(k^\times)^2$ 双射参数化。该构造用普通椭圆曲线和不变微分描述。我们确定短轨道和分歧、非分歧循环商以及中心对合下的商。对于 $q=8$，给出了在 $\F_2$ 上的显式平面模型。
英文摘要
Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case \textup{(ib)} in the classification of Giulietti--Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera $g=2^h+1$ with $h\ge2$. For $g\ge9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge8$, the isomorphism classes in case \textup{(ib)} of genus $q+1$ are parametrized bijectively by $(k^\times)^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over $\F_2$ is given.
