Serre's modularity conjecture

The Serre 's conjecture is a mathematical theorem on Galois representations and modular forms, which was proved in 2006 by Chandrashekhar Khare, Jean -Pierre Wintenberger and Mark Kisin. The Serre 's conjecture implies the Modularitätssatz and thus also the great Fermat's theorem. The Serre 's conjecture goes back to a conjecture of Serre.

Regardless of Khare and Wintenberger also demonstrated Luis Dieulefait 2004 special cases of Serre 's conjecture, which are sufficient for the proof of the major theorem of Fermat.

Formulation

The conjecture relates Galois representations of the absolute Galois group of the rational numbers.

Be an absolutely irreducible, continuous and odd dimensional representation of over a finite field

The characteristic

According to the assumption there is a Hecke eigenform

Level, weight, and type of side

Such that for all primes, prime to the following shall apply:

And

The stage and the weight of explicitly calculated in Serres article.

It is already known for a long time by deep sets of Goro Shimura, Deligne, Barry Mazur and Robert Langlands, that ( as requested above) can assign each Hecke eigenform a presentation. The Serre 's conjecture asserts the converse: Every irreducible, continuous and odd appearance comes from a modular form.

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