Topology Seminar: Andrew Salch

In the 1970s, work of Adams, Baird, Bousfield, and Ravenel gave a description of the orders of the KU[1/2]-local stable homotopy groups of spheres as the denominators of special values of the Riemann zeta-function. Meanwhile, Lichtenbaum conjectured a formula, ultimately proven 30 years later as a consequence of the Iwasawa main theorem and the norm residue theorem, relating the orders of the algebraic K-groups of totally real number rings to special values of their Dedekind zeta-functions. In this talk I will describe two general approaches, an analytic approach and an algebraic approach, to a general kind of number theory that arises in any tensor triangulated category: this is a general framework for the above results and gneralizations of them, and which aims to describe the orders of Bousfield-localized stable homotopy groups of finite spectra in terms of special values of L-functions. Then I'll show off some new results in this framework, in particular, a "universal" description of the KU-local homotopy groups of the Moore spectrum S/p in terms of L-values, and as a consequence, a proof of a certain (infinite) family of cases of Leopoldt's conjecture, by counting orders of homotopy groups.