Modular symbols, Eisenstein series, and congruences

Jay Heumann; Vinayak Vatsal

doi:10.5802/jtnb.886

Jay Heumann ¹ ; Vinayak Vatsal ²

¹ University of Wisconsin-Stout 712 South Broadway Menomonie, WI 54751
² University of British Columbia 1984 Mathematics Road Vancouver V6T 1Z2, Canada

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

Abstract (VO)
Résumé

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k \geq 2$ and of the same level $N$ , both eigenfunctions of the Hecke operators, and both normalized so that $a_{1} (f) = a_{1} (E) = 1$ . 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 $\bar{ℚ}$ lying over a rational prime $p > 2$ ), the algebraic parts of the special values $L (E, χ, j)$ and $L (f, χ, j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,

\frac{τ (\bar{χ}) L (f, χ, j)}{{(2 π i)}^{j - 1} Ω_{f}^{sgn (E)}} \equiv \frac{τ (\bar{χ}) L (E, χ, j)}{{(2 π i)}^{j} Ω_{E}} (mod 𝔭)

where the sign of $E$ is $\pm 1$ depending on $E$ , and $Ω_{f}^{sgn (E)}$ is the corresponding canonical period for $f$ . Also, $χ$ is a primitive Dirichlet character of conductor $m$ , $τ (\bar{χ})$ is a Gauss sum, and $j$ is an integer with $0 < j < k$ such that ${(- 1)}^{j - 1} \cdot χ (- 1) = sgn (E)$ . Finally, $Ω_{E}$ is a $𝔭$ -adic unit which is independent of $χ$ 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.

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 $a_{1} (f) = a_{1} (E) = 1$ . Le résultat principal de cet article est le suivant : si $E$ et $f$ sont congruents modulo un idéal premier $𝔭 ∣ p$ , alors les valeurs spéciales des fonctions $L (E, χ, j)$ et $L (f, χ, j)$ sont également congruentes modulo $𝔭$ . Plus précisement, on montre que

\frac{τ (\bar{χ}) L (f, χ, j)}{{(2 π i)}^{j - 1} Ω_{f}^{sgn (E)}} \equiv \frac{τ (\bar{χ}) L (E, χ, j)}{{(2 π i)}^{j} Ω_{E}} (mod 𝔭)

où le signe $sgn (E)$ est $\pm 1$ et ne dépend que de $E$ , et $Ω_{f}^{sgn (E)}$ est la période canonique de $f$ . Ici $χ$ désigne un caractère primitif de Dirichlet de conducteur $m$ , $τ (\bar{χ})$ une somme de Gauss, et $j$ un entier tel que $0 < j < k$ et ${(- 1)}^{j - 1} \cdot χ (- 1) = sgn (E)$ . Enfin, $Ω_{E}$ est une unité $𝔭$ -adique indépendante de $χ$ 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.

MR

DOI : 10.5802/jtnb.886

Affiliations des auteurs :

Jay Heumann ¹ ; Vinayak Vatsal ²

¹ University of Wisconsin-Stout 712 South Broadway Menomonie, WI 54751
² University of British Columbia 1984 Mathematics Road Vancouver V6T 1Z2, Canada

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     title = {Modular symbols, {Eisenstein} series, and congruences},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Jay Heumann; Vinayak Vatsal. Modular symbols, Eisenstein series, and congruences. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 709-756. doi : 10.5802/jtnb.886. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.886/

Bibliographie
Cité par

[1] J. Bellaïche, and S. Dasgupta, The $p$ -adic L-functions of evil Eisenstein series, preprint, (2012).

[2] J.E. Cremona, Algorithms for Modular Elliptic Curves, Second ed. Cambridge: Cambridge University Press, (1997). | MR | Zbl

[3] F. Diamond and J. Im, Modular Forms and Modular Curves, Conference Proceedings, Canadian Math. Soc., 17, (1995), 39–133. | MR | Zbl

[4] E. Friedman, Ideal class groups in basic $Z_{p_{1}} \times \dots \times Z_{p_{s}}$ -extensions of abelian number fields, Invent. Math., 65, (1981/82), 425–440. | MR | Zbl

[5] R. Greenberg and G. Stevens, $p$ -adic $L$ -functions and $p$ -adic Periods of Modular Forms, Invent. Math. 111, (1993), 407–447. | MR | Zbl

[6] H. Hida, Elementary Theory of $L$ -functions and Eisenstein series, Cambridge: Cambridge University Press, (1993). | MR | Zbl

[7] H. Hida, Galois representations into $G l_{2} (Z_{p} [[X]]$ associated to ordinary cusp forms, Invent. Math. 85, (1985), 545-613. | MR | Zbl

[8] Y. Hirano, Congruences of modular forms and the Iwasawa $λ$ -invariants, preprint, (2014). | MR

[9] B. Mazur, On the Arithmetic of Special Values of $L$ -functions, Invent. Math. 55, (1979), 207–240. | MR | Zbl

[10] T. Miyake, Modular Forms, New York: Springer-Verlag, (1989). | Zbl

[11] H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory I. Classical Theory, New York: Cambridge University Press, (2007). | MR | Zbl

[12] A.A. Ogg, Modular Forms and Dirichlet Series, New York: W.A. Benjamin Inc., (1969). | MR | Zbl

[13] H. Rademacher, Topics in Analytic Number Theory, New York: Springer-Verlag, (1973). | MR | Zbl

[14] B. Schoeneberg, Elliptic Modular Functions, New York: Springer-Verlag, (1974). | MR | Zbl

[15] W.A. Stein, Modular Forms, a Computational Approach, Providence, RI: American Mathematical Society, (2007). | MR | Zbl

[16] G. Stevens, Arithmetic on Modular Curves, Boston: Birkhauser, (1982). | MR | Zbl

[17] G. Stevens, The Eisenstein Measure and Real Quadratic Fields, Theorie des Nombres, Quebec, (1989), 887–927. | MR | Zbl

[18] V. Vatsal, Canonical Periods and Congruence Formulae, Duke Math. J., 98, 2, (1999), 397–419. | MR | Zbl

[19] A. Wiles, Modular elliptic curves and Fermat’s last theorem., Annals of Mathematics, 141, (1995), 443–551. | MR | Zbl

Dominique Bernardi; Bernadette Perrin-Riou Modular symbols and Petersson's product, Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 795-859 | DOI:10.5802/jtnb.1143 | Zbl:1468.11098
Yuichi Hirano Congruences of modular forms and the Iwasawa $λ$ -invariants, Bulletin de la Société Mathématique de France, Volume 146 (2018) no. 1, pp. 1-79 | Zbl:1446.11076

Cité par 2 documents. Sources : zbMATH