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Ring-theoretic properties of certain Hecke algebras. (English) Zbl 0823.11030

This paper provides a key fact needed in the previous paper by A. Wiles [Ann. Math. (2) 141, No. 3, 443–551 (1995; Zbl 0823.11029); namely that (certain) Hecke algebras are complete intersections. The key ideas are a computation of Euler characteristics (Tate-Poitou) and the introduction of auxiliary primes p1modp n such that the projective limit (as n) of the corresponding Hecke algebras is a power series ring. To see this one estimates its minimal number of generators (say d) and then shows that its dimension is at least d+1. To derive the assertion one uses that the original Hecke algebra can be defined by dividing this projective limit by d equations. It should be noted that this treats only the minimal case, where the condition “minimal” is used in the computation of Euler characteristics.

An appendix explains a remark of the reviewer which allows one to simplify some arguments. The authors were probably too exhausted to find this additional shortcut.


MSC:
11G05Elliptic curves over global fields
11F11Holomorphic modular forms of integral weight
11D41Higher degree diophantine equations
13C40Linkage, complete intersections and determinantal ideals
14M10Complete intersections
14H52Elliptic curves