亏格$4$的超特殊Howe曲线的计数算法
Algorithm to enumerate superspecial Howe curves of genus $4$
摘要：
Howe曲线是作为两条椭圆曲线的$\mathbf{P}^1$上的纤维积而获得的亏格$4$的曲线。任何Howe曲线都是正则的。本文给出了一个求超特殊Howe曲线及其同构类的有效算法。本文不仅讨论了Howe曲线的超特殊性的判定算法，而且还讨论了Howe曲线的同构判定算法。我们的算法是更有效的比传统的作者提出的一般典型曲线到目前为止。通过在计算机代数系统Magma上执行我们的算法，证明了特征$7<p\le 331$中超特殊Howe曲线的存在性，并列举了特征$p\le 53$中超特殊Howe曲线的同构类.
英文摘要：
A Howe curve is a curve of genus $4$ obtained as the fiber product over $\mathbf{P}^1$ of two elliptic curves. Any Howe curve is canonical. This paper provides an efficient algorithm to find superspecial Howe curves and that to enumerate their isomorphism classes. We discuss not only an algorithm to test the superspeciality but also an algorithm to test isomorphisms for Howe curves. Our algorithms are much more efficient than conventional ones proposed by the authors so far for general canonical curves. We show the existence of a superspecial Howe curve in characteristic $7<p\le 331$ and enumerate the isomorphism classes of superspecial Howe curves in characteristic $p\le 53$, by executing our algorithms over the computer algebra system Magma.
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