Generalized Kummer congruences and $p$-adic families of motives

Abstract

We describe some new general constructions of $p$-adic $L$-functions attached to certain arithmetically defined complex $L$-functions coming from motives over $\bold Q$ with coefficiens in a number field $T$, with $[T:\bold Q]<\infty$. These constructions are equivalent to proving some generalized Kummer congruences for critical special values of these complex $L$-functions.