MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 453 common ps which is given in Chapter 5.Believing now that the proof was complete,I sketched the whole theory in three lectures in Cambridge,England on June 21-23.However,it became clear to me in the fall of 1993 that the con- struction of the Euler system used to extend Flach's method was incomplete and possibly fawed. Chapter 3 follows the original approach I had taken to the problem of bounding the Selmer group but had abandoned on learning of Flach's paper Darmon encouraged me in February,1994,to explain the reduction to the com- plete intersection property,as it gave a quick way to exhibit infinite families of modular j-invariants.In presenting it in a lecture at Princeton I mad almost unconsciously,a critical switch to the special primes used in Chapter 3 as auxiliary primes.I had only observed the existence and importance of these primes in the fall of 1992 while trying to extend Flach's work.Previously,I had only used primes q≡ -1modp as auxiliary primes.In hindsight this change was crucial because of a development due to de Shalit.As explained before,I had realized earlier that Hida's theory often provided one step towards a power series ring at least in the ordinary case. At the Cambridge conference de Shali had explained to me that for primes g=1 modp he had obtained a version of Hida's results.But except for explaining the complete intersection argument in the lecture at Princeton I still did not give any thought to my initial ap proach,which I had put aside since the summer of 1991,since I continued to believe that the Euler system approach was the correct one. Meanwhile in January,1994,R.Taylor had joined me in the attempt to repair the Euler system argument.Then in the spring of 1994,frustrated in the efforts to repair the Euler system argument,I began to work with Taylor on an attempt to devise a new argument u =2 reached an impasse at the end of August.As Taylor was still not convinced that the Euler system argument was irreparable,I decided in September to take one last look at my attempt to generalise Flach,if only to formulate more precisely the obstruction.In doing this I came suddenly to a marvelous revelation: saw in a flash on September 19th,1994,that de Shalit's theory,if generalised, could be used together with duality to glue the Hecke rings at suitable auxiliary levels into a power series ring.I had unexpectedly found the missing key tomy old abandoned approach.It was the old idea of picking gi's with gi =1 mod p" and nioo as ioo that I used to achieve the limiting process.The switch to th special primes of Chapter3 had made all this possible After I communicated the argument to Taylor,we spent the next few days making sure of the details.The full argument,together with the deduction of the e complete inters ction property,is given in [TW] In conclusion the key breakthrough in the proof had been the realization in the spring of 1991 that the two invariants introduced in the appendix could be used to relate the deformation rings and the Hecke rings.In effect the n-MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 453 common p~,which is given in Chapter 5. Believing now that the proof was complete, I sketched the whole theory in three lectures in Cambridge, England on June 21-23. However, it became clear to me in the fall of 1993 that the con￾struction of the Euler system used to extend Flach's method was incomplete and possibly flawed. Chapter 3 follows the original approach I had taken to the problem of bounding the Selmer group but had abandoned on learning of Flach's paper. Darmon encouraged me in February, 1994, to explain the reduction to the com￾plete intersection property, as it gave a quick way to exhibit infinite families of modular j-invariants. In presenting it in a lecture at Princeton, I made, almost unconsciously, a critical switch to the special primes used in Chapter 3 as auxiliary primes. I had only observed the existence and importance of these primes in the fall of 1992 while trying to extend Flach's work. Previously, I had only used primes q - -1 modp as auxiliary primes. In hindsight this change was crucial because of a development due to de Shalit. As explained before, I had realized earlier that Hida's theory often provided one step towards a power series ring at least in the ordinary case. At the Cambridge conference de Shalit had explained to me that for primes q E 1modp he had obtained a version of Hida's results. But except for explaining the complete intersection argument in the lecture at Princeton, I still did not give any thought to my initial ap￾proach, which I had put aside since the summer of 1991, since I continued to believe that the Euler system approach was the correct one. Meanwhile in January, 1994, R. Taylor had joined me in the attempt to repair the Euler system argument. Then in the spring of 1994, frustrated in the efforts to repair the Euler system argument, I began to work with Taylor on an attempt to devise a new argument using p = 2. The attempt to use p = 2 reached an impasse at the end of August. As Taylor was still not convinced that the Euler system argument was irreparable, I decided in September to take one last look at my attempt to generalise Flach, if only to formulate more precisely the obstruction. In doing this I came suddenly to a marvelous revelation: I saw in a flash on September 19th, 1994, that de Shalit's theory, if generalised, could be used together with duality to glue the Hecke rings at suitable auxiliary levels into a power series ring. I had unexpectedly found the missing key to my old abandoned approach. It was the old idea of picking qi7s with qi = 1modpn" and n, -+ oo as i -+ oo that I used to achieve the limiting process. The switch to the special primes of Chapter 3 had made all this possible. After I communicated the argument to Taylor, we spent the next few days making sure of the details. The full argument, together with the deduction of the complete intersection property, is given in [TW]. In conclusion the key breakthrough in the proof had been the realization in the spring of 1991 that the two invariants introduced in the appendix could be used to relate the deformation rings and the Hecke rings. In effect the 7-