On Generalised Kummer Congruences and Higher Rank Iwasawa Theory at Arbitrary Weights

Abstract

In recent work Burns, Kurihara and Sano introduced a natural notion of (generalised) Stark elements of arbitrary rank and weight and conjectured a precise congruence relation between Stark elements of a fixed rank and different weights. This conjecture was shown to simultaneously generalise both the classical congruences of Kummer and the explicit reciprocity law of Artin-Hasse and Iwasawa. In this article, we show that the congruence conjecture also implies that the Iwasawa theoretical `zeta element’ that is conjecturally associated to the multiplicative group has good interpolation properties at arbitrary even integers.