A combinatorial interpretation of Serre's conjecture on modular Galois representations

Herremans, Adriaan

Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1287-1321.

On donne une conjecture concernant les représentations modulaires de $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ 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.

[1] H. Cohen; J. Oesterlé Dimensions des espaces de formes modulaires (Lecture Notes in Mathematics), Volume 627 (1977), pp. 69-78 | Zbl

[2] H. Darmon Serre's conjectures, Seminar on Fermat's Last Theorem (Toronto, ON, 1993-1994) (CMS Conf. Proc.), Volume 17, pp. 135-153 | Zbl

[3] P. Deligne; J.-P. Serre Formes modulaires de poids, Ann. Sci. E.N.S, Volume 7 (1974), pp. 507-530 | EuDML | Numdam | MR | Zbl

[4] F. Diamond; J. Im Modular forms and modular curves, Seminar on Fermat's Last Theorem (CMS conference proceedings) (1995), pp. 39-134 | Zbl

[5] B. Edixhoven The weight in Serre's conjectures on modular forms, Inventiones Mathematicae, Volume 109 (1992), pp. 563-594 | DOI | EuDML | MR | Zbl

[6] B. Edixhoven Serre's conjecture, Modular Forms and Fermat's Last Theorem (1997), pp. 209-242 | Zbl

[7] A. Herremans A combinatorial interpretation of Serre's conjecture on modular Galois representations (2001) Ph.D.thesis K.U.Leuven (28th May)

[8] A. Knapp Elliptic Curves, Oxford University Press, 1992 | MR | Zbl

[9] J. Manin Parabolic Points and Zeta-Function of Modular Curves, Math. USSR Izvestija, Volume 6 (1972), pp. 19-64 | DOI | MR | Zbl

[10] H. Matsumura Commutative algebra, W. A. Benjamin, New York, 1970 | MR | Zbl

[11] F. Martin Périodes de formes modulaires de poids 1 (2001) Thèse de doctorat Paris 7 (20 décembre)

[12] L. Merel Universal Fourier expansions of modular forms (Lecture Notes in Mathematics), Volume 1585 (1994), pp. 59-94 | Zbl

[13] T. Miyake Modular Forms, Springer-Verlag, 1989 | MR | Zbl

[14] J.-P. Serre Sur les représentations modulaires de degré 2 de $\mathrm{Gal}(\overline{\mathbb{Q}/\mathbb{Q})}$, Duke Mathematical Journal, Volume 54 (1987) no. 1, pp. 179-230 | DOI | MR | Zbl

[15] J.-P. Serre Oeuvres, collected papers, Volume vol. III (1986), pp. 1972-1984 | Zbl

[16] G. Shimura Introduction to the Arithmetic Theory of Automorphic Functions, Iwana Shoten Publishers and Princeton University Press, 1971 | MR | Zbl

[17] V. Shokurov Shimura integrals of cusp forms, Math. USSR Isvestija, Volume 16 (1981), pp. 603-646 | DOI | Zbl