Bloch groups, algebraic K-theory, units, and Nahm's Conjecture

2017/12/13 by Frank Calegari, Calegari, Frank, Stavros Garoufalidis +3 · 3 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1712.04887

openalex publication_date 2017/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an element of the Bloch group of a number field~F and a natural number~n, we construct an explicit unit in the field F_n=F(e^{2 πi/n}), well-defined up to \nn-th powers of nonzero elements of~F_n. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~F with the K-group K₃(F) gives \changed(up to an unidentified invertible scalar) a \changedformula for a certain abstract Chern class from~K₃(F). The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove Nahm's conjecture relating the modularity of certain q-hypergeometric series to the vanishing of the associated elements in the Bloch group of~\mathbb Q.

Cited by

• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Modularity from q-series
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples

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