p-adic J-homomorphisms and a product formula

Abstract

One-point compactification turns real vector spaces into spheres. In homotopy theory, this transformation gets encoded in a map called the "real J-homomorphism". Here we define and investigate p-adic J-homomorphisms, which sort of turn p-adic vector spaces into spheres. Our main theorem is a product formula for these J-homomorphisms, saying what happens when you start with a rational vector space. This formula specializes to Hilbert's version of the quadratic reciprocity law after applying second homotopy groups.

Added Conjecture 4.5 on alternative description of J_{F_p}, otherwise just minor edits

References in corpus (6)

• Higher Topos Theory
• Infinite-dimensional vector bundles in algebraic geometry (an introduction)
• Units of ring spectra and Thom spectra
• Stably dualizable groups
• Algebraic K-theory of the fraction field of topological K-theory
• Geometric construction of metaplectic covers of $\GL_{n}$ in characteristic zero

Cited by in corpus (5)

• A simple universal property of Thom ring spectra
• Brauer groups and Galois cohomology of commutative ring spectra
• A K-theoretic approach to Artin maps
• Reciprocity laws and $K$-theory
• Adjoining roots in homotopy theory