Modular symbols, Eisenstein series, and congruences

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 (2014)

  • Volume: 26, Issue: 3, page 709-756
  • ISSN: 1246-7405

Abstract

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Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k ≥ 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 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, τ ( χ ¯ ) L ( f , χ , j ) ( 2 π i ) j - 1 Ω f sgn ( E ) ≡ τ ( χ ¯ ) L ( E , χ , j ) ( 2 π i ) j Ω E ( mod 𝔭 ) where the sign of E is ± 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 ( χ ¯ ) is a Gauss sum, and j is an integer with 0 < j < k such that ( - 1 ) j - 1 · 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.

How to cite

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Heumann, Jay, and Vatsal, Vinayak. "Modular symbols, Eisenstein series, and congruences." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 709-756. <http://eudml.org/doc/275808>.

@article{Heumann2014,
abstract = {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 $a_1(f)=a_1(E) = 1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $\mathfrak\{p\}$ (which we take in this paper to be a prime of $\overline\{\mathbb\{Q\}\}$ lying over a rational prime $p &gt;2$), 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,\[ \frac\{\tau (\bar\{\chi \})L(f,\chi ,j)\}\{(2 \pi i)^\{j-1\}\Omega \_f^\{\text\{sgn\}(E)\}\} \equiv \frac\{\tau (\bar\{\chi \})L(E,\chi ,j)\}\{(2 \pi i)^\{j\}\Omega \_E\} \pmod \{\mathfrak\{p\}\} \]where the sign of $E$ is $\pm 1$ depending on $E$, and $\Omega _f^\{\text\{sgn\}(E)\}$ is the corresponding canonical period for $f$. Also, $\chi $ is a primitive Dirichlet character of conductor $m$, $\tau (\bar\{\chi \})$ is a Gauss sum, and $j$ is an integer with $0&lt; j&lt; k$ such that $(-1)^\{j-1\}\cdot \chi (-1) = \text\{sgn\}(E)$. Finally, $\Omega _E$ is a $\mathfrak\{p\}$-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.},
affiliation = {University of Wisconsin-Stout 712 South Broadway Menomonie, WI 54751; University of British Columbia 1984 Mathematics Road Vancouver V6T 1Z2, Canada},
author = {Heumann, Jay, Vatsal, Vinayak},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {12},
number = {3},
pages = {709-756},
publisher = {Société Arithmétique de Bordeaux},
title = {Modular symbols, Eisenstein series, and congruences},
url = {http://eudml.org/doc/275808},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Heumann, Jay
AU - Vatsal, Vinayak
TI - Modular symbols, Eisenstein series, and congruences
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 709
EP - 756
AB - 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 $a_1(f)=a_1(E) = 1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $\mathfrak{p}$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p &gt;2$), 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,\[ \frac{\tau (\bar{\chi })L(f,\chi ,j)}{(2 \pi i)^{j-1}\Omega _f^{\text{sgn}(E)}} \equiv \frac{\tau (\bar{\chi })L(E,\chi ,j)}{(2 \pi i)^{j}\Omega _E} \pmod {\mathfrak{p}} \]where the sign of $E$ is $\pm 1$ depending on $E$, and $\Omega _f^{\text{sgn}(E)}$ is the corresponding canonical period for $f$. Also, $\chi $ is a primitive Dirichlet character of conductor $m$, $\tau (\bar{\chi })$ is a Gauss sum, and $j$ is an integer with $0&lt; j&lt; k$ such that $(-1)^{j-1}\cdot \chi (-1) = \text{sgn}(E)$. Finally, $\Omega _E$ is a $\mathfrak{p}$-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.
LA - eng
UR - http://eudml.org/doc/275808
ER -

References

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