Modular functions and resolvent problems

The link between modular functions and algebraic was a driving force behind the 19th century study of both. Examples include solutions by Hermite Klein quintic via elliptic general sextic level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques circle “resolvent problems” formulated pursued Klein, Hilbert others. As one example, we prove that essential dimension at $$p=2$$ for symmetric groups $$S_n$$ is equal certain -coverings defined using moduli spaces principally polarized abelian varieties. Our proofs use deformation theory varieties in characteristic p, specifically Serre-Tate theory, as well family remarkable mod symplectic -representations constructed Jordan. shown an appendix Nate Harman, properties need such representations exist only case. In second half this introduce notion $$\mathcal {E}$$ -versality kind generalization Kummer many congruence covers are -versal. We these result deduce equivalence Hilbert’s 13th Problem (and related conjectures) with problems about covers.