The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arithmetic objects (motives). In recent years general conjectures [B-K] have been formulated relating these special values to the Selmer groups and other arithmetic invariants of the associated motives. In view of these conjectures, congruences between special values should give certain congruences between orders of the corresponding Selmer groups.
In this paper, we take a different point of view by studying congruences between Selmer groups and deducing consequences of such congruences to the congruences of special values. Roughly speaking, given two Galois representations that are congruent on a finite level, i.e., that become isomorphic representations modulo a prime power, we consider the relation between the corresponding Selmer groups. Precise definitions will be given in later sections. We will focus on the Selmer groups defined by Bloch and Kato [B K] and will provide congruences when the congruent Galois representations are from cyclotomic characters, Hecke characters from CM elliptic curves, and from adjoints of modular forms.
Methods used in this paper are mostly from Iwasawa theory, including the classical theory originated from Iwasawa [Iw] and the ``horizontal'' theory recently developed by Wiles [Wi], even though only the algebraic part of the theory is applied.
There are several versions of Selmer groups, motivated by the Selmer groups of elliptic curves. We choose to study the Selmer group of Bloch and Kato since it is the one that applies to the most general context and