Computing Stark units for totally real cubic fields

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark’s conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) an...

Stark's Conjectures and Hilbert's Twelfth Problem

We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples. In a series of important papers [Stark 71, S...

Gross–Stark units for totally real number fields

Explicit Computation of Gross-stark Units over Real Quadratic Fields

We present an effective and practical algorithm for computing Gross-Stark units over a real quadratic base field F. Our algorithm allows us to explicitly construct certain relative abelian extensions of F where these units lie, using only information from the base field. These units were recently proved to always exist within the correct extension fields of F by Dasgupta, Darmon, and Pollack, w...

On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields, when they are ordered by their absolute discriminants. For an order O in a cubic field, we study the three groups: Cl2(O), the group of ideal classes of O of order 2; Cl2 (O), the group of narr...