The 600-Cell in the Millennium Problems: Poincaré's Dodecahedral Space, the Icosahedral Curve (BSD), and Hodge (INDYNA Research Note 37)

Abstract

The objects of the INDYNA formula — the 120 icosians (the vertices of the 600-cell), E₈, the icosahedron and the field ℚ(√5) — appear exactly in several of the seven Millennium problems. We map where, and carry out the explicit computations that the register's objects allow: the Laplace spectrum of Poincaré's dodecahedral space from the icosian characters, its E₈ plumbing and Seifert invariants; point counts of Bring's icosahedral curve and the special values of the associated elliptic L-function, which give analytic rank 0 over ℚ and over ℚ(√5) and hence the weak Birch–Swinnerton-Dyer statement for this curve; and the double appearance of E₈ in the K3 lattice. No Millennium problem is solved here.