Arbitrary Conjugations of CM Types

This chapter is based on an unpublished article of Tate [Ta], who formulated a conjecture extending the fundamental theorem of complex multiplication to the case when the automorphism σ does not leave the reflex field fixed. Tate obtains a commutative diagram just as before, up to an idele of square 1, thus leaving the conjecture that this idele can in fact be taken to be 1. This conjecture is equivalent to an important special case of a conjecture of Langlands concerning the conjugation of Shimura varieties [Lglds]. Tate reformulates the conjecture in terms of a “type transfer”. The first two sections of the chapter give the general algebraic number theory setting for this type transfer, and the final sections give the application to the abelian varieties with complex multiplication.