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Zbl 0823.11029
Wiles, Andrew
Modular elliptic curves and Fermat's Last Theorem. (English)
[J] Ann. Math. (2) 141, No. 3, 443-551 (1995). ISSN 0003-486X; ISSN 1939-0980/e

The main result is the proof of the Taniyama-Weil conjecture for a large class of elliptic curves over $\Bbb Q$. These include semistable curves, and thus the result implies the famous Fermat conjecture. \par To achieve this one shows that in many cases the Hecke algebra of a modular curve is the base of a universal deformation of the associated $p$-adic Galois representation. Here $p\geq 3$, and the representation modulo $p$ must be irreducible. If this holds for $p=3$, then everything follows from results of Langlands-Tunnell, as $\text {PGL} (2,\Bbb F\sb 3) \cong S\sb 4$ is solvable. If the mod 3 representation is reducible, one can use $p=5$ (and the result for $p=3$). \par In the meantime there has been more progress, extending the result to elliptic curves with semistable reduction at 3 and 5. The restriction stems from the argument above, and limitations of our present crystalline techniques. \par The contents in more detail. \par Chapter I introduces the universal deformation ring, various local conditions on representations, and the corresponding tangent spaces. These are $H\sp 1$'s of certain cohomology theories, and the corresponding Euler characteristic is computed using the results of Tate- Poitou. \par Chapter II treats Hecke algebras. It is shown that they are Gorenstein (this comes down to multiplicity one), and Ribet's theory of change of level is used to start the reduction to the minimal case. A key fact is always that certain Hecke operators are redundant in the definition of the Hecke algebra. This is easy for primes of good reduction, but involved for the others. \par Chapter III brings the introduction of certain auxiliary primes $q\equiv 1\bmod p\sp n$ which are also very important in the subsequent paper of Taylor-Wiles [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)]. It then reduces the assertion to the fact that the Hecke algebra is a complete intersection. That this condition holds is the content of Taylor-Wiles. \par Chapter IV treats the dihedral case. This does not occur for semistable curves, and requires the techniques of Kolyvagin-Rubin. \par Chapter V actually proves the Taniyama-Weil conjecture for many elliptic curves. \par An appendix explains the relevant commutative algebra.
[G.Faltings (Bonn)]

MSC 2000:: *11G05 Elliptic curves over global fields
11F11 Modular forms, one variable
11D41 Higher degree diophantine equations
14H52 Elliptic curves

Keywords: Fermat's last theorem; proof of the Taniyama-Weil conjecture; elliptic curves; semistable curves; Hecke algebra of a modular curve; $p$-adic Galois representation

Citations: Zbl 0823.11030

Cited in: Zbl 1210.11115 Zbl 1225.11076 Zbl 1169.11020 Zbl 1193.11055 Zbl 1122.11030 Zbl 1121.11063 Zbl 1230.11055 Zbl 1081.11035 Zbl 1062.11022 Zbl 1173.11328 Zbl 1121.11045 Zbl 1053.11048 Zbl 1040.11043 Zbl 1064.11026 Zbl 1057.11032 Zbl 1051.11031 Zbl 1042.11031 Zbl 1031.11031 Zbl 1028.11036 Zbl 1026.11035 Zbl 1076.11035 Zbl 1062.11034 Zbl 0998.11030 Zbl 1097.14033 Zbl 0983.00001 Zbl 0982.11033 Zbl 1027.11086 Zbl 1017.11500 Zbl 1007.11001 Zbl 0977.60025 Zbl 0952.11014 Zbl 1009.11034 Zbl 1005.11030 Zbl 0930.00002 Zbl 0899.11051 Zbl 1015.11019 Zbl 0919.11041 Zbl 0916.11017 Zbl 0914.11030 Zbl 0912.11024 Zbl 0908.11017 Zbl 0903.13003 Zbl 0957.11028 Zbl 0957.11027 Zbl 0919.11042 Zbl 0882.11001 Zbl 0878.11017 Zbl 0867.11032 Zbl 0864.13009 Zbl 0855.11057 Zbl 1027.11502 Zbl 0976.11028 Zbl 0921.11029 Zbl 0877.11035 Zbl 0867.11041 Zbl 0864.11031 Zbl 0864.11029 Zbl 0843.11001 Zbl 0840.11012 Zbl 0826.11014 Zbl 0824.00025 Zbl 0823.11030 Zbl 0924.11046 Zbl 0830.11015 Zbl 0824.14017

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