Journal Article

HIGHER ORDER CONGRUENCES AMONGST HASSE–WEIL -VALUES

PETERS, LLOYD, DELBOURGO, DANIEL

2015

Overview

For the$(d+1)$-dimensional Lie group$G=\\mathbb{Z}_{p}^{\\times }\\ltimes \\mathbb{Z}_{p}^{\\oplus d}$, we determine through the use of$p$-power congruences a necessary and sufficient set of conditions whereby a collection of abelian$L$-functions arises from an element in$K_{1}(\\mathbb{Z}_{p}\\unicode[STIX]{x27E6}G\\unicode[STIX]{x27E7})$. If$E$is a semistable elliptic curve over$\\mathbb{Q}$, these abelian$L$-functions already exist; therefore, one can obtain many new families of higher order$p$-adic congruences. The first layer congruences are then verified computationally in a variety of cases.

• Publisher: Cambridge University Press
• Subject: Congruences