Modular symbols, Eisenstein series, and congruences

Heumann, Jay; Vatsal, Vinayak

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 709-756.

Soient $E$ une série d’Eisenstein et $f$ une forme modulaire parabolique, de même niveau $N$. Supposons que $E$ et $f$ soient vecteurs propres pour les opérateurs de Hecke, et qu’ils soient tous les deux normalisés de sorte que $a_1\left(f\right)=a_1\left(E\right)=1$. Le résultat principal de cet article est le suivant : si $E$ et $f$ sont congruents modulo un idéal premier $𝔭\mid p$, alors les valeurs spéciales des fonctions $L(E,\chi ,j)$ et $L(f,\chi ,j)$ sont également congruentes modulo $𝔭$. Plus précisement, on montre que

$$\frac{\tau \left(\overline{\chi }\right)L(f,\chi ,j)}{{\left(2\pi i\right)}^{j-1}{\Omega }_f^{\text{sgn}\left(E\right)}}\equiv \frac{\tau \left(\overline{\chi }\right)L(E,\chi ,j)}{{\left(2\pi i\right)}^j{\Omega }_E}(mod𝔭)$$

où le signe $\text{sgn}\left(E\right)$ est $\pm 1$ et ne dépend que de $E$, et ${\Omega }_f^{\text{sgn}\left(E\right)}$ est la période canonique de $f$. Ici $\chi$ désigne un caractère primitif de Dirichlet de conducteur $m$, $\tau \left(\overline{\chi }\right)$ une somme de Gauss, et $j$ un entier tel que $0<j<k$ et ${(-1)}^{j-1}·\chi (-1)=\text{sgn}\left(E\right)$. Enfin, ${\Omega }_E$ est une unité $𝔭$-adique indépendante de $\chi$ et de $j$. Ce résultat est une généralisation des travaux de Stevens et Vatsal en poids $k=2$.

Dans cet article on construit le symbole modulaire de $E$, et on calcule les valeurs spéciales. La dernière section conclut avec des exemples numériques du théorème principal.

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p>2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,

$$\frac{\tau \left(\overline{\chi }\right)L(f,\chi ,j)}{{\left(2\pi i\right)}^{j-1}{\Omega }_f^{\text{sgn}\left(E\right)}}\equiv \frac{\tau \left(\overline{\chi }\right)L(E,\chi ,j)}{{\left(2\pi i\right)}^j{\Omega }_E}(mod𝔭)$$

where the sign of $E$ is $\pm 1$ depending on $E$, and ${\Omega }_f^{\text{sgn}\left(E\right)}$ is the corresponding canonical period for $f$. Also, $\chi$ is a primitive Dirichlet character of conductor $m$, $\tau \left(\overline{\chi }\right)$ is a Gauss sum, and $j$ is an integer with $0<j<k$ such that ${(-1)}^{j-1}·\chi (-1)=\text{sgn}\left(E\right)$. Finally, ${\Omega }_E$ is a $𝔭$-adic unit which is independent of $\chi$ and $j$. This is a generalization of earlier results of Stevens and Vatsal for weight $k=2$.

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.

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