Higher order congruences amongst Hasse-Weil L-values

For the (d+1)-dimensional Lie group G=Z× pZp ⊕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 K1ZpG. 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.