On Stickelberger Elements for $\mathbb{Q}({\zeta }_{p^{n+1}}{)}^{+}$ and $p$-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.