We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4]. This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein orders. In addition we show that the endomorphism rings of certain principally polarized, absolutely simple, and ordinary abelian surfaces are not computable in subexponential time with current methods [4], [21], including ours.

中文翻译：

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阿贝尔曲面自同形环的计算

我们扩展了 Bisson [4] 的结果，以计算基本极化、绝对简单和普通阿贝尔表面的自同态环，这些表面在基场大小的次指数时间内定义在有限域上。此处覆盖的阿贝尔曲面在 [4] 中被排除在外。这是通过使用 [17] 中介绍的技术有效地确定 Bass 和 Gorenstein 阶次的超阶来实现的。此外，我们表明某些主要极化的、绝对简单的和普通的阿贝尔表面的自同态环不能用当前的方法 [4]，[21]，包括我们的方法在次指数时间内计算。