Theorem of Taniyama-Shimura

The theorem of Taniyama-Shimura establishes an important connection between the elliptic curved , which are objects of the algebraic Géométrie, and the modular forms, which are certain holomorphic functions periodic studied in Théorie of the numbers. In spite of the name, which comes from the Conjecture of Taniyama-Shimura, the theorem is the work of Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

If p is a Prime number and E an elliptic curve on $\ mathbb {Q} \,$ (the body of the rational numbers), we can reduce the equation defining E modulo p ; for all the values, but in a finished way, p we will obtain an elliptic curve on the Corps finished $\ mathbb {F} _p \,$ , with $n_p \,$ elements. One can then consider the progression

$a_p = n_p - p \,$ ,

who is an important invariant of the elliptic curve E . Each modular form gives also a progression of numbers, by a Transformation of Fourier. An elliptic curve whose progression is in agreement with that obtained starting from a modular form is called modular . The theorem of Taniyama-Shimura establishes this:

"All the elliptic curves on $\ mathbb {Q} \,$ are modulaires."

This Théorème was conjectured in first by Yutaka Taniyama in September 1955. With Goro Shimura, it improved of it the rigor until in 1957. Taniyama died in 1958. In the Sixties, the theorem became associated with the Programme with Langlands with unification with the conjectures in mathematics, and was consequently component a key. The conjecture was recovered and promoted by André Weil in the Seventies, and the name of Weil was sometimes associated with this theorem. Some estimated however that it was indémontrable.

In the Eighties when Gerhard Frey suggested that the conjecture of Taniyama-Shimura (such as it was called then) implied the Dernier theorem of Fermat. It did it while trying to show that any counterexample of the last theorem of Fermat would lead to a not-modular elliptic curve. Ken Ribet showed this result later. In 1995, Andrew Wiles and Richard Taylor showed a particular case of the theorem of Taniyama-Shimura (the case of the semi-stable elliptic curved ), and this demonstration was sufficiently strong to provide the proof of the last theorem of Fermat.

The complete theorem of Taniyama-Shimura was finally shown in 1999 by Breuil, Conrad, Diamond and Taylor which, being based on the work of Wiles, covered by jumps of chip the remaining cases until obtaining the demonstration of the complete result.

Several theorems of the theory of the numbers similar to the last theorem of Fermat rise from the theorem of Taniyama-Shimura. For example: " no cube can be written like a sum of two numbers Premiers between them high with the power N with N ≥ 3" (the case N = 3 was already known of Euler).

In March 1996, Wiles divided the Prix Wolf with Robert Langlands. Although none of both showed the theorem, it was recognized that they had established the key results which made it possible to build its demonstration.

References

Henri Darmon: has Proof off the Full Shimura-Taniyama-Weil Conjecture Is Announced , Notices off the American Mathematical Society, vol. 46 (1999), No 11. A pleasant introduction of the theorem and the broad outlines of the demonstration contains (in English).
Brian Conrad, Fred Diamond, Richard Taylor: Modularity off some potentially Barsotti-Touches Welsh representations , Journal off the American Mathematical Society 12 (1999), pp. 521-567. The demonstration contains.

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