*(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 |