Modular functions and resolvent problems: With an appendix by Nate Harman
Modular functions and resolvent problems: With an appendix by Nate Harman
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模块化功能和已解决的问题：内特·哈曼 (Nate Harman) 的附录
DOI：
10.1007/s00208-022-02395-8
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发表时间：
2022
影响因子：
1.4
通讯作者：
Wolfson, Jesse
中科院分区：
文献类型：
--
作者：
Farb, Benson;Kisin, Mark;Wolfson, Jesse
模函数和代数函数之间的联系是世纪对两者研究的推动力。例子包括解决方案的厄米和克莱因的五次通过椭圆模函数和一般的六次通过水平2超椭圆函数。本文旨在将现代算术技术应用于克莱因、希尔伯特等人所提出和追求的“预解问题”的循环。作为一个例子，我们证明了对称群的本质维数等于用主极化交换簇的模空间定义的某些覆盖的本质维数。我们的证明使用变形理论的阿贝尔品种的characteristicp，特别是塞尔泰特理论，以及家庭的显着模2辛表示约旦建造。正如Nate Harman在附录中所示，我们需要的这种表示的性质只存在于这种情况下。在本文的后半部分，我们引入了-versality的概念作为库默理论的一种推广，并证明了许多同余覆盖是-versality的。利用这些结果，我们证明了Hilbert第13问题（及相关问题）与同余覆盖问题的等价性。
The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of “resolvent problems” formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension atfor the symmetric groupsis equal to the essential dimension at 2 of certain-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristicp, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in thecase. In the second half of this paper we introduce the notion of-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are-versal. We use these-versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers.