Verifying a p-adic abelian Stark conjecture at s=1

In a previous paper (Ann. L’ Inst. Fourier 52(2) (2002) 379–417) the second-named author developed a new approach to the abelian p-adic Stark conjecture at s=1 and stated some related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these—the ‘Weak Refined Combined Conjecture’—in 15 cases.