Geometry of the Wiman–Edge pencil and the Wiman curve

Abstract

The Wiman-Edge pencil is the universal family C t , t ∈ B of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group A 5 . The curve C 0 , discovered by Wiman in 1895 [11] and called the Wiman curve, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group S 5 . In this paper we give an explicit uniformization of B as a non-congruence quotient Γ\H of the hyperbolic plane H, where Γ < PSL 2 (Z) is a subgroup of index 18. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of C t into 10 lines (resp. 5 conics) whose intersection graph is the Petersen graph (resp. K 5 ).In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve C 0 itself as the quotient Λ\H, where Λ is a principal level 5 subgroup of a certain "unit spinor norm" group of Möbius transformations. We then prove that C 0 is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

Arithmeticity of the monodromy of the Wiman–Edge pencil

“…In other words, the involution ι lifts in an anti-linear manner to V o . As proved in Theorem 1.1 (see also Remark 2.5) of [7], there is an identification B • = Γ\H with Γ ⊂ PSL 2 (Z) being torsion free and so V o pulls back to H as a trivial symplectic local system with Γ-action. The basis (v, v ) of V o (C o ) constructed in Proposition 3.8 extends to one of the pull-back of V o to H (so we use c o as our base point).…”

“…It is elementary to see that the cycle type of these generators is (3, 2) for τ 6 , (4) for τ 4 (so both are odd) and (2, 2) for τ 2 (so τ 2 is even) and that τ 6 τ 4 τ 2 = 1; see [7, §2.3]. In fact, Theorem 2.1 of [7] implies that any ordered triple (τ 6 , τ 4 , τ 2 ) of generators S 5 whose orders are as their subscript and satisfy τ 6 τ 4 τ 2 = 1, differ from the triple above by an inner automorphism. So any such triple comes from some choice of K. We shall exploit this below.…”

Arithmeticity of the monodromy of the Wiman-Edge pencil

“…10, 6 closed geodesics that are pairwise disjoint. As shown in §2 of [7], these A 5 -orbits make up a configuration of K 5 -type resp. dodecahedral type.…”

“…The stabilizer of such a point is cyclic and the orientation of C o singles out a natural generator τ j (counter clockwise rotation over 2π/j around p j ). It is elementary to see that the cycle type of these generators is (3, 2) for τ 6 , (4) for τ 4 (so both are odd) and (2, 2) for τ 2 (so τ 2 is even) and that τ 6 τ 4 τ 2 = 1; see §2.3 of [7]. In fact, Theorem 2.1 of [7] implies that any ordered triple (τ 6 , τ 4 , τ 2 ) of generators S 5 whose orders are as their subscript and satisfy τ 6 τ 4 τ 2 = 1, differ from the triple above by an inner automorphism.…”

“…It is elementary to see that the cycle type of these generators is (3, 2) for τ 6 , (4) for τ 4 (so both are odd) and (2, 2) for τ 2 (so τ 2 is even) and that τ 6 τ 4 τ 2 = 1; see §2.3 of [7]. In fact, Theorem 2.1 of [7] implies that any ordered triple (τ 6 , τ 4 , τ 2 ) of generators S 5 whose orders are as their subscript and satisfy τ 6 τ 4 τ 2 = 1, differ from the triple above by an inner automorphism. So any such triple comes from some choice of K. We shall exploit this below.…”