Modularity and uniformization of a higher genus algebraic space curve, its distinct arithmetical realizations by cohomology groups and $E_6$, $E_7$, $E_8$-singularities

Abstract

We prove the modularity for an algebraic space curve Y of genus 50 in P 5 , which consists of 21 quartic polynomials in six variables, by means of an explicit modular parameterizations by theta constants of order 13. This provides an example of modularity for higher genus space curve as well as an explicit uniformization of algebraic space curves of higher genus and a hyperbolic uniformization of arithmetic type for a higher genus arithmetic algebraic curve. In particular, it gives a new solution for Hilbert's 22nd problem, i.e., uniformization of analytic relations by means of automorphic functions, by means of an explicit construction of uniformity for an overdetermined system of algebraic relations among five variables by theta constants. This gives 21 modular equations of order 13, which greatly improve the result of Ramanujan and Evans on the construction of modular equations of order 13. We show that this curve Y is isomorphic to the modular curve X(13). The corresponding ideal I(Y ) is invariant under the action of SL(2, 13), which leads to a new (21-dimensional, complementary series) representation of SL(2, 13). We determine both a small set of generators for a polynomial defining ideal and the minimum number of equations needed to define this algebraic space curve Y in P 5 . The projection Y → Y /SL(2, 13) (identified with CP 1 ) is a Galois covering whose generic fibre is interpreted as the Galois resolvent of the modular equation Φ 13 (•, j) = 0 of level 13, i.e., the function field of Y is the splitting field of this modular equation over C(j). The ring of invariant polynomials (C[z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ]/I(Y )) SL(2,13) over the modular curve X(13) leads to a new perspective on the theory of E 6 , E 7 and E 8 -singularities. Contents1. Introduction 2. Modularity for equations of E 6 , E 7 and E 8 -singularities coming from modular curves X(3), X(4) and X(5) 3. Invariant theory and modular forms for SL(2, 13) I: the quadratic and cubic invariants 4. Invariant theory and modular forms for SL(2, 13) II:the quartic invariants and the modular curve X(13) 5. An invariant ideal defining the modular curve X(13) 6. A new representation (complementary series) of SL(2, 13) 7. Modularity for an invariant ideal and an explicit uniformization of algebraic space curves of higher genus 8. Galois covering Y → Y /SL(2, 13) ∼ = CP 1 , Galois resolvent for the modular equation of level 13 and a Hauptmodul for Γ 0 (13) 9. Invariant theory and modular forms for SL(2, 13) III: some computation for invariant polynomials 10. Modularity for equations of E 6 , E 7 and E 8 -singularities coming from C Y /SL(2, 13) and variations of E 6 , E 7 and E 8 -singularity structures over X(13) 11. Modularity for equations of Q 18 and E 20 -singularities coming from C Y /SL(2, 13) and variations of Q 18 and E 20 -singularity structures over X(13)