Some remarks on p-adic L-functions for certain abelian varieties with complex multiplication

Abstract

Let [Formula: see text] be a quadratic field with prime discriminant [Formula: see text]. Let [Formula: see text] be an odd prime which splits in [Formula: see text]. In this paper, by generalizing our previous method in Iwasawa theory at the prime [Formula: see text], we construct a [Formula: see text]-adic canonical measure for the abelian varieties [Formula: see text] coming from the restriction of scalars to [Formula: see text] of quadratic twists of Gross curves. We then prove an elliptic analogue of Iwasawa’s theorem for the measure. As an application, we prove some cases of the [Formula: see text]-part Buhler–Gross conjecture for [Formula: see text] in the same spirit of Rubin.