Construction of elliptic $\mathfrak{p}$-units

TL;DR

This paper generalizes a method for constructing elliptic $rak{p}$-units in abelian extensions of imaginary quadratic fields, expressing their $p$-adic valuations via elliptic units, extending previous split prime cases to non-split scenarios.

Contribution

It extends Solomon's construction of elliptic $rak{p}$-units to non-split primes, providing explicit valuation formulas in terms of elliptic units.

Findings

Constructed pairs of elliptic $rak{p}$-units depending on Iwasawa algebra generators.

Expressed $p$-adic valuations of $rak{p}$-units using elliptic units' $p$-adic logarithm.

Computed constant terms of Coleman power series using recent results of T. Seiriki.

Abstract

Let $L/k$ be a finite abelian extension of an imaginary quadratic number field $k$. Let $\mathfrak{p}$ denote a prime ideal of $\mathcal{O}_k$ lying over the rational prime $p$. We assume that $\mathfrak{p}$ splits completely in $L/k$ and that $p$ does not divide the class number of $k$. If $p$ is split in $k/\mathbb{Q}$ the first named author has adapted a construction of Solomon to obtain elliptic $\mathfrak{p}$-units in $L$. In this paper we generalize this construction to the non-split case and obtain in this way a pair of elliptic $\mathfrak{p}$-units depending on a choice of generators of a certain Iwasawa algebra (which here is of rank 2). In our main result we express the $\mathfrak{p}$-adic valuations of these $\mathfrak{p}$-units in terms of the $p$-adic logarithm of an explicit elliptic unit. The crucial input for the proof of our main result is the computation of the constant term of a suitable Coleman power series, where we rely on recent work of T. Seiriki.