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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 $$p\equiv 1\bmod p^ n$$ such that the projective limit (as $$n\to \infty$$) 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.
Reviewer: G.Faltings (Bonn)

##### MSC:
 11G05 Elliptic curves over global fields 11F11 Holomorphic modular forms of integral weight 11D41 Higher degree equations; Fermat’s equation 13C40 Linkage, complete intersections and determinantal ideals 14M10 Complete intersections 14H52 Elliptic curves
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