Real regulator maps with finite 0-locus

TL;DR

This paper studies families of algebraic curves derived from Laurent polynomials with cyclotomic edge polynomials, proving finiteness properties of certain loci related to regulator maps and normal functions.

Contribution

It establishes the finiteness of the real split locus of a regulator extension and related torsion and A-polynomial loci for these curve families.

Findings

The real split locus of the regulator extension is finite.

The torsion locus of the normal function is finite.

The A-polynomial locus for the family of curves is finite.

Abstract

A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus $g$ curves carrying a canonical algebraic $K_2$-class over a $g$-dimensional base $S$, hence to an extension of admissible variations of MHS (or normal function) on $S$. We prove that the $\mathbb{R}$-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the $A$-polynomial locus for the family of curves are also finite.