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 , by their counterparts in the theory of modular symbols.
On donne une conjecture concernant les représentations modulaires de 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 , par leurs équivalents dans la théorie des symboles modulaires modulo .
Keywords: modular forms, modular symbols, 2-dimensional irreducible Galois representations, Shimura cohomology
Mot clés : formes modulaires, symboles modulaires, représentations galoisiennes irréductibles, cohomologie de Shimura
Herremans, Adriaan 1
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TY - JOUR AU - Herremans, Adriaan TI - A combinatorial interpretation of Serre's conjecture on modular Galois representations JO - Annales de l'Institut Fourier PY - 2003 SP - 1287 EP - 1321 VL - 53 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1980/ DO - 10.5802/aif.1980 LA - en ID - AIF_2003__53_5_1287_0 ER -
%0 Journal Article %A Herremans, Adriaan %T A combinatorial interpretation of Serre's conjecture on modular Galois representations %J Annales de l'Institut Fourier %D 2003 %P 1287-1321 %V 53 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1980/ %R 10.5802/aif.1980 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1980/
[1] Dimensions des espaces de formes modulaires (Lecture Notes in Mathematics), Volume 627 (1977), pp. 69-78 | Zbl
[2] Serre's conjectures, Seminar on Fermat's Last Theorem (Toronto, ON, 1993-1994) (CMS Conf. Proc.), Volume 17, pp. 135-153 | Zbl
[3] Formes modulaires de poids, Ann. Sci. E.N.S, Volume 7 (1974), pp. 507-530 | EuDML | Numdam | MR | Zbl
[4] Modular forms and modular curves, Seminar on Fermat's Last Theorem (CMS conference proceedings) (1995), pp. 39-134 | Zbl
[5] The weight in Serre's conjectures on modular forms, Inventiones Mathematicae, Volume 109 (1992), pp. 563-594 | DOI | EuDML | MR | Zbl
[6] Serre's conjecture, Modular Forms and Fermat's Last Theorem (1997), pp. 209-242 | Zbl
[7] A combinatorial interpretation of Serre's conjecture on modular Galois representations (2001) Ph.D.thesis K.U.Leuven (28th May)
[8] Elliptic Curves, Oxford University Press, 1992 | MR | Zbl
[9] Parabolic Points and Zeta-Function of Modular Curves, Math. USSR Izvestija, Volume 6 (1972), pp. 19-64 | DOI | MR | Zbl
[10] Commutative algebra, W. A. Benjamin, New York, 1970 | MR | Zbl
[11] Périodes de formes modulaires de poids 1 (2001) Thèse de doctorat Paris 7 (20 décembre)
[12] Universal Fourier expansions of modular forms (Lecture Notes in Mathematics), Volume 1585 (1994), pp. 59-94 | Zbl
[13] Modular Forms, Springer-Verlag, 1989 | MR | Zbl
[14] Sur les représentations modulaires de degré 2 de , Duke Mathematical Journal, Volume 54 (1987) no. 1, pp. 179-230 | DOI | MR | Zbl
[15] Oeuvres, collected papers, Volume vol. III (1986), pp. 1972-1984 | Zbl
[16] Introduction to the Arithmetic Theory of Automorphic Functions, Iwana Shoten Publishers and Princeton University Press, 1971 | MR | Zbl
[17] Shimura integrals of cusp forms, Math. USSR Isvestija, Volume 16 (1981), pp. 603-646 | DOI | Zbl
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