Let and be an Eisenstein series and a cusp form, respectively, of the same weight and of the same level , both eigenfunctions of the Hecke operators, and both normalized so that . The main result we prove is that when and are congruent mod a prime (which we take in this paper to be a prime of lying over a rational prime ), the algebraic parts of the special values and satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,
where the sign of is depending on , and is the corresponding canonical period for . Also, is a primitive Dirichlet character of conductor , is a Gauss sum, and is an integer with such that . Finally, is a -adic unit which is independent of and . This is a generalization of earlier results of Stevens and Vatsal for weight .
In this paper we construct the modular symbol attached to an Eisenstein series, and compute the special values. We give numerical examples of the congruence theorem stated above, and in the penultimate section we give the proof of the congruence theorem.
Soient une série d’Eisenstein et une forme modulaire parabolique, de même niveau . Supposons que et soient vecteurs propres pour les opérateurs de Hecke, et qu’ils soient tous les deux normalisés de sorte que . Le résultat principal de cet article est le suivant : si et sont congruents modulo un idéal premier , alors les valeurs spéciales des fonctions et sont également congruentes modulo . Plus précisement, on montre que
où le signe est et ne dépend que de , et est la période canonique de . Ici désigne un caractère primitif de Dirichlet de conducteur , une somme de Gauss, et un entier tel que et . Enfin, est une unité -adique indépendante de et de . Ce résultat est une généralisation des travaux de Stevens et Vatsal en poids .
Dans cet article on construit le symbole modulaire de , et on calcule les valeurs spéciales. La dernière section conclut avec des exemples numériques du théorème principal.
@article{JTNB_2014__26_3_709_0, author = {Jay Heumann and Vinayak Vatsal}, title = {Modular symbols, {Eisenstein} series, and congruences}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {709--756}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {3}, year = {2014}, doi = {10.5802/jtnb.886}, mrnumber = {3320499}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.886/} }
TY - JOUR AU - Jay Heumann AU - Vinayak Vatsal TI - Modular symbols, Eisenstein series, and congruences JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 709 EP - 756 VL - 26 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.886/ DO - 10.5802/jtnb.886 LA - en ID - JTNB_2014__26_3_709_0 ER -
%0 Journal Article %A Jay Heumann %A Vinayak Vatsal %T Modular symbols, Eisenstein series, and congruences %J Journal de théorie des nombres de Bordeaux %D 2014 %P 709-756 %V 26 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.886/ %R 10.5802/jtnb.886 %G en %F JTNB_2014__26_3_709_0
Jay Heumann; Vinayak Vatsal. Modular symbols, Eisenstein series, and congruences. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 709-756. doi : 10.5802/jtnb.886. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.886/
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