A combinatorial interpretation of Serre's conjecture on modular Galois representations
Annales de l'Institut Fourier, Volume 53 (2003) no. 5, p. 1287-1321

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.

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.

DOI : https://doi.org/10.5802/aif.1980
Classification:  11F11,  11F25,  11F30,  11F67,  11F75,  11F80,  11R32
Keywords: modular forms, modular symbols, 2-dimensional irreducible Galois representations, Shimura cohomology
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {5},
     year = {2003},
     pages = {1287-1321},
     doi = {10.5802/aif.1980},
     zbl = {1056.11032},
     mrnumber = {2032935},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_5_1287_0}
}
Herremans, Adriaan. A combinatorial interpretation of Serre's conjecture on modular Galois representations. Annales de l'Institut Fourier, Volume 53 (2003) no. 5, pp. 1287-1321. doi : 10.5802/aif.1980. http://www.numdam.org/item/AIF_2003__53_5_1287_0/

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