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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 $p\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{n}$ 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.

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