450 ANDREW WILES a new technique in order to deal with the trivial zeroes.It involved replacing the standard Iwasawa theory method of considering the fields in the cyclotomic Z-extension by a similar analysis based on a choice of infinitely many distinct Some aspects of this method suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of Wr,might be to make a comparison between the cohomology groups as varies but with the field Q fixed.The new principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones.After reading the paper [Grel],I realized that the duality theorems in Galois cohomology of Poitou and Tate would be useful for this.The crucial extract tfrom this latter theory is in Section 2 of Chapter In order to put these ideas into practice I developed in a naive form the techniques of the first two sections of Chapter 2.This drew in particular on a detailed study of all the c ngruen ces betw en f and other m dular form of differing levels,a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first cohomology group well under two assumptions,first that a certain subgroup of the second cohomology vanished and sond that the form choeat the minimal level These assumptions were much too restrictive to be really effective but at least they pointed in the right direction.Some of these arguments are to be found in the second section of Chapter 1 and some form the first weak approximatior to the argument in Chapter 3.At that time.however.I used auxiliary primes g=-1 modp when varying E as the geometric techniques I worked with did mot apply in general for mes g=1modp (This was as for much the sam reason that the reduction of level argument in Ril]is much more difficult when g=1modp.)In all this work I used the more general assumption that Pp was modular rather than the assumption that p=3. In the late 1980's,I translated these ideas into ring-theoretic language.A few years previously Hida had constructed some explicit one-parameter fam- ilies of Galois representations.In an attempt to understand this.Mazur had been developing the language of Galois representati More- over,Mazur realized that the universal deformation rings he found should be given by Hecke rings,at least in certain special cases.This critical conjecture refined the exp ation that all ordinary liftings of modular repre ntations should be modular.In making the translation to this ring-theoretic language I realized that the vanishing assumption on the subgroup of H2 which I had needed should be replaced by the stronger condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one could estimate the number of generators and relations and so made the original assumption more plausible. To be of use,the defo rmation th ory required some development.Apart from some special examples examined by Boston and Mazur there had been 450 ANDREW WILES a new technique in order to deal with the trivial zeroes. It involved replacing the standard Iwasawa theory method of considering the fields in the cyclotomic Zp-extension by a similar analysis based on a choice of infinitely many distinct primes qi r 1modpnt with ni -+ co as i -t co. Some aspects of this method suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of Wf, might be to make a comparison between the cohomology groups as C varies but with the field Q fixed. The new principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones. After reading the paper [Grel], I realized that the duality theorems in Galois cohomology of Poitou and Tate would be useful for this. The crucial extract from this latter theory is in Section 2 of Chapter 1. In order to put these ideas into practice I developed in a naive form the techniques of the first two sections of Chapter 2. This drew in particular on a detailed study of all the congruences between f and other modular forms of differing levels, a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first cohomology group well under two assumptions, first that a certain subgroup of the second cohomology group vanished and second that the form f was chosen at the minimal level for m. These assumptions were much too restrictive to be really effective but at least they pointed in the right direction. Some of these arguments are to be found in the second section of Chapter 1and some form the first weak approximation to the argument in Chapter 3. At that time, however, I used auxiliary primes q - -1 modp when varying C as the geometric techniques I worked with did not apply in general for primes q - lmodp. (This was for much the same reason that the reduction of level argument in [Rill is much more difficult when q E 1modp.) In all this work I used the more general assumption that pp was modular rather than the assumption that p = 3. In the late 1980's, I translated these ideas into ring-theoretic language. A few years previously Hida had constructed some explicit one-parameter fam￾ilies of Galois representations. In an attempt to understand this, Mazur had been developing the language of deformations of Galois representations. More￾over, Mazur realized that the universal deformation rings he found should be given by Hecke rings, at least in certain special cases. This critical conjecture refined the expectation that all ordinary liftings of modular representations should be modular. In making the translation to this ring-theoretic language I realized that the vanishing assumption on the subgroup of H2 which I had needed should be replaced by the stronger condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one could estimate the number of generators and relations and so made the original assumption more plausible. To be of use, the deformation theory required some development. Apart from some special examples examined by Boston and Mazur there had been