This paper describes a generalisation of the methods of Iwasawa Theory to the field obtained by adjoining the field of definition of all the -power torsion points on an elliptic curve, , to a number field, . Everything considered is essentially well-known in the case has complex multiplication, thus it is assumed throughout that has no complex multiplication. Let denote the Galois group of over . Then the main focus of this paper is on the study of the -cohomology of the Selmer group of over , and the calculation of its Euler characteristic, where possible. The paper also describes proposed natural analogues to this situation of the classical Iwasawa -invariant and the condition of having -invariant equal to 0.
The final section illustrates the general theory by a detailed discussion of the three elliptic curves of conductor 11, at the prime .
"Euler characteristics and elliptic curves II." J. Math. Soc. Japan 53 (1) 175 - 235, January, 2001. https://doi.org/10.2969/jmsj/05310175