Lagrange Resolvents Constructed from Stark Units

Abstract

Classical Gauss sums are Lagrange resolvents formed from the Gaussian periods lying in a cyclic extension K over ℚ of prime conductor. Elliptic Gauss sums and elliptic resolvents (which are particular instances of Lagrange resolvents) play an important role in the theory of abelian extensions of imaginary quadratic fields. Motivated by the close relationship between the Stark units and Gaussian periods in a cyclic extension K ⊂ ℚ(ζ p ) and the analogies between Stark units over totally real fields and elliptic units over imaginary quadratic fields, we consider for the first time Lagrange resolvents constructed from Stark units over totally real fields and study the differences and similarities they share with classical Gauss sums.