Real regulator maps with finite 0-locus

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

-class over a g -dimensional base S , hence to an extension of admissible variations of mixed Hodge structure (or normal function) on S . We prove that the

double struck upper R R $\mathbb{R}$

-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the A -factor locus for the family of curves are also finite.