Modular functions and resolvent problems

TL;DR

This paper explores the connection between modular functions and algebraic functions, applying modern arithmetic techniques to classical resolvent problems, and introduces the concept of E-versality to relate Hilbert's 13th Problem to congruence covers.

Contribution

It applies modern deformation theory and introduces E-versality to advance understanding of resolvent problems and their relation to Hilbert's 13th Problem.

Findings

Essential dimension at p=2 for S_n matches that of certain S_n-coverings.

Proves E-versality of many congruence covers.

Establishes equivalence of Hilbert's 13th Problem with problems about congruence covers.

Abstract

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 at $p=2$ for the symmetric groups $S_n$ is equal to the essential dimension at $2$ of certain $S_n$-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic $p$, specifically Serre-Tate theory, as well as a family of remarkable mod $2$ symplectic $S_n$-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the $p=2$ case. In the second half of this paper we introduce the notion of $\E$-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are $\E$-versal. We use these $\E$-versality result to deduce the equivalence of Hilbert's 13th Problem (and related conjectures) with problems about congruence covers.