# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Modular elliptic curves and Fermat’s Last Theorem. (English) Zbl 0823.11029

The main result is the proof of the Taniyama-Weil conjecture for a large class of elliptic curves over $ℚ$. These include semistable curves, and thus the result implies the famous Fermat conjecture.

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 $p\ge 3$, and the representation modulo $p$ must be irreducible. If this holds for $p=3$, then everything follows from results of Langlands-Tunnell, as $\text{PGL}\left(2,{𝔽}_{3}\right)\cong {S}_{4}$ is solvable. If the mod 3 representation is reducible, one can use $p=5$ (and the result for $p=3$).

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.

The contents in more detail.

Chapter I introduces the universal deformation ring, various local conditions on representations, and the corresponding tangent spaces. These are ${H}^{1}$’s of certain cohomology theories, and the corresponding Euler characteristic is computed using the results of Tate- Poitou.

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.

Chapter III brings the introduction of certain auxiliary primes $q\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}{p}^{n}$ 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.

Chapter IV treats the dihedral case. This does not occur for semistable curves, and requires the techniques of Kolyvagin-Rubin.

Chapter V actually proves the Taniyama-Weil conjecture for many elliptic curves.

An appendix explains the relevant commutative algebra.

##### MSC:
 11G05 Elliptic curves over global fields 11F11 Holomorphic modular forms of integral weight 11D41 Higher degree diophantine equations 14H52 Elliptic curves