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首页> 外文期刊>Canadian Journal of Mathematics >The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians
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The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians

机译：（4,4）-拆分雅可比矩阵的两条曲线的算术

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摘要

In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.
机译：在本文中，我们研究了属2曲线，其雅可比矩阵承认极化（4，4）-同质为椭圆曲线的乘积。我们认为特征场不同于2和3，我们不认为它们是代数封闭的。我们获得了所有主要极化的阿贝尔曲面的完整分类，这些曲面可以通过将两个椭圆曲线沿其4个方向胶合而产生，并且可以推导出它们的绝对不变性满足的关系。作为中间步骤，我们对属2曲线的Jacobian之间的Richelot等位基因进行了一般性描述，其中以前仅考虑了具有在基域上逐点定义的内核的Richelot等位基因。我们的主要工具是2类曲线的Galois理论表征，允许多个Richelot同构。

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