HIGHER ORDER CONGRUENCES AMONGST HASSE-WEIL L-VALUES

For the (d + 1)-dimensional Lie group G = Z(p)(x) x Z(p)(circle plus d), we determine through the use of p-power congruences a necessary and su ffi cient set of conditions whereby a collection of abelian L-functions arises from an element in K-1(Z(p)[G]). If E is a semistable elliptic curve over 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.