Congruences between Selmer groups ∗

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 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. We also display consequences of such congruences to the congruences of special values of L-functions, in particular, to special values of the Riemann zeta function. It is interesting to observe that the congruences of special values obtained this way are different from the classical congruences, such as the Kummer congruences. Methods used in this paper are mostly from Iwasawa theory, including the classical theory originated from Iwasawa [Iw] and the ”horizontal” theory recently developed