A combinatorial interpretation of Serre's conjecture on modular Galois representations
[Un équivalent combinatoire de la conjecture de Serre concernant les représentations modulaires galoisiennes]
Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1287-1321.

On donne une conjecture concernant les représentations modulaires de Gal ( ¯/) sur les corps finis qui est de nature combinatoire (sans utiliser de formes modulaires). On démontre que cette conjecture est équivalente à celle de Serre. L’idée principale est de remplacer les formes modulaires à coefficients dans un corps fini de caractéristique p, par leurs équivalents dans la théorie des symboles modulaires modulo p.

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic p, by their counterparts in the theory of modular symbols.

DOI : https://doi.org/10.5802/aif.1980
Classification : 11F11,  11F25,  11F30,  11F67,  11F75,  11F80,  11R32
Mots clés : formes modulaires, symboles modulaires, représentations galoisiennes irréductibles, cohomologie de Shimura
@article{AIF_2003__53_5_1287_0,
     author = {Herremans, Adriaan},
     title = {A combinatorial interpretation of Serre's conjecture on modular Galois representations},
     journal = {Annales de l'Institut Fourier},
     pages = {1287--1321},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {5},
     year = {2003},
     doi = {10.5802/aif.1980},
     zbl = {1056.11032},
     mrnumber = {2032935},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1980/}
}
Herremans, Adriaan. A combinatorial interpretation of Serre's conjecture on modular Galois representations. Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1287-1321. doi : 10.5802/aif.1980. http://www.numdam.org/articles/10.5802/aif.1980/

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