Modular functions and resolvent problems

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 $$\mathcal {E}$$ -versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are $$\mathcal {E}$$ -versal. We use these $$\mathcal {E}$$ -versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers.