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

Taylor, Richard; Wiles, Andrew

Ann. Math. (2) 141, No. 3, 553-572 (1995).

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

Keywords:

Galois representations; Fermat’s last theorem; Hecke algebras; complete intersections; Euler characteristics

Citations:

Zbl 0823.11029