On Stickelberger Elements for $\mathbb{Q}(\zeta_{p^{n+1}})^+$ and $p$-adic $L$-functions

TL;DR

This paper surveys known constructions of $p$-adic $L$-functions, introduces explicit 'real' Stickelberger elements in Galois group rings, and demonstrates their coherence and relation to twisted $p$-adic $L$-functions in cyclotomic towers.

Contribution

It constructs explicit 'real' Stickelberger elements that annihilate class groups and links them to twisted $p$-adic $L$-functions, extending Iwasawa's framework.

Findings

Construction of explicit 'real' Stickelberger elements.

Demonstration of coherence in $bZ_p$-towers.

Connection to twisted $p$-adic $L$-functions.

Abstract

We give a survey of a couple known constructions of $p$-adic $L$-functions including Iwasawa's construction from classical Stickelberger elements. We then construct "real" Stickelberger elements, i.e., explicit elements in the Galois group ring with $\mathbb{Z}_p$ coefficients that annihilate the Sylow $p$-subgroup of the ideal class group of $\mathbb{Q}(\zeta_{p^{n+1}})^+$. In analogy with Iwasawa's work, we show that these elements are coherent in $\mathbb{Z}_p$-towers and give rise to twisted $p$-adic $L$-functions.