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

Adriaan Herremans[1]

  • [1] University of Utrecht, Department of Mathematics, PO Box 80010, 3508 TA Utrecht (The Netherlands)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 5, page 1287-1321
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Herremans, Adriaan. "A combinatorial interpretation of Serre's conjecture on modular Galois representations." Annales de l’institut Fourier 53.5 (2003): 1287-1321. <http://eudml.org/doc/116073>.

@article{Herremans2003,
abstract = {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.},
affiliation = {University of Utrecht, Department of Mathematics, PO Box 80010, 3508 TA Utrecht (The Netherlands)},
author = {Herremans, Adriaan},
journal = {Annales de l’institut Fourier},
keywords = {modular forms; modular symbols; 2-dimensional irreducible Galois representations; Shimura cohomology; Eichler-Shimura-isomorphism; Serre conjecture; modular form; Hecke operator; modular symbol},
language = {eng},
number = {5},
pages = {1287-1321},
publisher = {Association des Annales de l'Institut Fourier},
title = {A combinatorial interpretation of Serre's conjecture on modular Galois representations},
url = {http://eudml.org/doc/116073},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 5
SP - 1287
EP - 1321
AB - 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.
LA - eng
KW - modular forms; modular symbols; 2-dimensional irreducible Galois representations; Shimura cohomology; Eichler-Shimura-isomorphism; Serre conjecture; modular form; Hecke operator; modular symbol
UR - http://eudml.org/doc/116073
ER -

References

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  11. F. Martin, Périodes de formes modulaires de poids 1, (2001) 
  12. L. Merel, Universal Fourier expansions of modular forms, 1585 (1994), 59-94, Springer-Verlag Zbl0844.11033
  13. T. Miyake, Modular Forms, (1989), Springer-Verlag Zbl0701.11014MR1021004
  14. J.-P. Serre, Sur les représentations modulaires de degré 2 de G a l ( ¯ Q / Q ) , Duke Mathematical Journal 54 (1987), 179-230 Zbl0641.10026MR885783
  15. J.-P. Serre, Oeuvres, collected papers vol. III (1986), 1972-1984, Springer-Verlag Zbl0849.01049
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  17. V. Shokurov, Shimura integrals of cusp forms, Math. USSR Isvestija 16 (1981), 603-646 Zbl0466.14014

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