A generalization of the Ross symbols in higher K-groups and hypergeometric functions I

TL;DR

This paper generalizes Ross symbols to higher K-groups for certain varieties and demonstrates that their Beilinson regulators can be expressed using hypergeometric functions, extending the understanding of algebraic K-theory and special functions.

Contribution

It introduces a new class of symbols in higher K-groups and links their regulators to hypergeometric functions, broadening the scope of algebraic K-theory and special functions.

Findings

Beilinson regulator described by hypergeometric functions {}_{d+3}F_{d+2}'s

Generalization of Ross symbols to higher K-groups of specific varieties

Establishment of a connection between algebraic K-theory and hypergeometric functions

Abstract

The Ross symbol is defined to be an element {1-z,1-w\} in K_2 of a Fermat curve z^n+w^m=1. Ross showed that it is non-torsion by computing the Beilinson regulator. In this paper, we introduce a generalization of the Ross symbols in K_{d+1} of a variety (1-x_0^{n_0})\cdots(1-x_d^{n_d})=t. The main result is that the Beilinson regulator is described by the hypergeometric functions {}_{d+3}F_{d+2}'s.