论文标题
固定$ 3 $ torsion的Abelian表面
Abelian surfaces with fixed $3$-torsion
论文作者
论文摘要
给定一个属两曲线$ x：y^2 = x^5 + a x^3 + b x^2 + c x + d $，我们对所有其他此类曲线$ y $进行明确的参数化，并在雅各布族人三个扭曲的雅各布人$ \ mbox {jac}（jac}（x）（x）[3] [3] $ con \ mbox（y）$ \ mbox（y）$ \ mbox（y）$ \ mbox {众所周知，在某些条件下，$ x $的模块化意味着无限的许多$ y $的模块化，我们解释了我们的公式如何使模块化的转移。我们的方法集中在复杂反射组的不变理论上，$ C_3 \ times \ operatatorName {sp} _4（\ Mathbf {f} _3）$。我们讨论了其他示例，其中复杂的反射组与曲线的模量空间有关，特别是通过阐述了组$ \ perperatorname {sp} _2（\ Mathbf {f} _3 _3）= \ slrm {sl} _2（umth} _2（$ alth）$ ____________________________________________________________________________________________3（曲线。
Given a genus two curve $X: y^2 = x^5 + a x^3 + b x^2 + c x + d$, we give an explicit parametrization of all other such curves $Y$ with a specified symplectic isomorphism on three-torsion of Jacobians $\mbox{Jac}(X)[3] \cong \mbox{Jac}(Y)[3]$. It is known that under certain conditions modularity of $X$ implies modularity of infinitely many of the $Y$, and we explain how our formulas render this transfer of modularity explicit. Our method centers on the invariant theory of the complex reflection group $C_3 \times \operatorname{Sp}_4(\mathbf{F}_3)$. We discuss other examples where complex reflection groups are related to moduli spaces of curves, and in particular motivate our main computation with an exposition of the simpler case of the group $\operatorname{Sp}_2(\mathbf{F}_3) = \mathrm{SL}_2(\mathbf{F}_3)$ and $3$-torsion on elliptic curves.
