Stark's conjecture in multi-quadratic extensions, revisited

Abstract

Stark’s conjectures connect special units in number fields with special values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions attached to these fields. We consider the fundamental equality of Stark’s refined conjecture for the case of an abelian Galois group, and prove it when this group has exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> . For biquadratic extensions and most others, we prove more, establishing the conjecture in full.