Modular symbols, Eisenstein series, and congruences

Heumann, Jay; Vatsal, Vinayak

doi:10.5802/jtnb.886

Cited by 4 publications

(12 citation statements)

References 14 publications

(37 reference statements)

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“…This is also true if F = Q and N = N is prime ([Qua11, Thm 3.6], relying on [Eme02]). Our main evidence for believing (3.3) holds more generally is that it seems necessary to obtain a congruence of L-values for newforms f congruent to E as in [HV14]. This is because a congruence of L-values would say that (up to constants) we have |P 0 K (φ)| 2 ≡ |P 0 K (φ 0 )| 2 mod p for infinitely many K. On the other hand, the map Cl(o K ) → Cl(O) appears to behave essentially randomly, so an infinitude of such congruences would seem to happen with probability 0 if cφ ≡ φ 0 .…”

Section: Quadratic Twist L-valuesmentioning

confidence: 79%

“…When F = Q, p > 2, and p ∤ N = N, one can deduce stronger nonvanishing results from Theorem 2.1 using [HV14].…”

Section: Quadratic Twist L-valuesmentioning

confidence: 96%

“…For instance, if χ is trivial, then L(1, E 2,N , χ) is essentially L(0, η K )L(1, η K ), which is essentially h 2 K . Vatsal and Heumann [Vat99], [HV14] show that if f is congruent to E 2,N mod p r , p > 2 and p ∤ N , then there is a congruence of special values between algebraic parts of L(1, f, χ) and L(1, E 2,N , χ). In fact they treat higher weights, other Eisenstein series and non-central special values (when k ≥ 4).…”

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confidence: 99%

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The Jacquet–Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight $2$

Martin

2017

Mathematical Research Letters

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We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular forms.A celebrated result of Mazur says that, for N a prime and p a prime dividing (the numerator of) N −1 12 , there exists a cusp form f ∈ S 2 (N ) = S 2 (Γ 0 (N )) congruent to the Eisenstein series E 2,N of weight 2 and level N mod p [Maz77, II(5.12)]. Further, if p = 2, one has a congruence for the algebraic part of the centralwhere η K is the quadratic character associated to a quadratic field K/Q [Maz79]. For instance, if N = 11 and K is not split at 11, there is one cusp form f ∈ S 2 (N ) and one gets that L alg (1, f K ) ≡ 0 mod 5 if and only if 5 ∤ h K . (If K splits at 11, the root number is −1 so L(1, f K ) = 0). A form of Mazur's L-value result was reproved by Gross [Gro87] for K/Q imaginary quadratic using quaternion algebras and the height pairing, whereas Mazur used modular symbols. Ramakrishnan pointed out to me that one can also deduce this from his average L-value formula with Michel [MR12]. (Note Gross's argument also involves an averaging type procedure, so these two arguments are not entirely different in spirit.)In this article, we use the Jacquet-Langlands correspondence and an explicit L-value formula to extend these results of Mazur both to more general levels and to parallel weight 2 Hilbert modular forms over a totally real field F with K/F a quadratic CM extension. For simplicity, we only state our results precisely for F = Q in this introduction.We first discuss the Hecke eigenvalue congruence result and a nonvanishing L-value result, and will state the more precise result on L-value congruences below in Theorem B. Theorem A. Let N be a nonsquare, and write N = N 1 N 2 where (N 1 , N 2 ) = 1 and N 1 has an odd number of prime factors, all of which occur to odd exponents. Let p be a prime dividing (the numerator of ) 1 12 ϕ(N 1 )N 2 q|N 2 (1 + q −1 ) and p a prime of Q above p. Then

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Section: Quadratic Twist L-valuesmentioning

confidence: 79%

“…When F = Q, p > 2, and p ∤ N = N, one can deduce stronger nonvanishing results from Theorem 2.1 using [HV14].…”

Section: Quadratic Twist L-valuesmentioning

confidence: 96%

mentioning

confidence: 99%

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The Jacquet–Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight $2$

Martin

2017

Mathematical Research Letters

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“…Remarquons d'abord que t étant dans le demi-plan de Poincaré, −it appartient au plan complexe privé de la demi-droite x ≤ 0, ce qui permet de donner un sens à (−it) s pour s ∈ C. En 0, F (t) est O(t −k ), donc l'intégrale est 2 On donne ainsi un sens à l'intégrale i∞ 0 F (t)(tx + y) k−2 dt qui ne converge pas en 0. Il est dans l'esprit des démonstrations des équations fonctionnelles des fonctions L (voir [6]). convergente en 0 pour Re(s) > k − 1.…”

Section: Ces éQuations Déterminent Entièrement Cunclassified

“…Remarque 1.12. Comme cela est remarqué dans [6], cette formule peut aussi s'écrire en remplaçant i par τ ∈ H :…”

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Symboles modulaires et produit de Petersson

Bernardi

Perrin-Riou

2021

Journal De Théorie Des Nombres De Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 795-859 Symboles modulaires et produit de Petersson par Dominique BERNARDI et Bernadette PERRIN-RIOU Résumé. On revisite des articles de Eichler et de Shimura afin de donner une formule algébrique (basée sur les symboles de Farey) pour le produit d'intersection sur l'espace des symboles modulaires tel qu'il est décrit par Pollack et Stevens. On définit l'homomorphisme de périodes d'une série d'Eisenstein (symbole d'Eisenstein-Dedekind-Stevens) et on étend le produit d'intersection à ces objets. On construit une base adaptée à un traitement algorithmique de l'espace des séries d'Eisenstein de période rationnelle pour Γ 0 (N ). On donne un algorithme pour construire un symbole de Farey d'un sousgroupe d'indice fini d'un groupe donné par un symbole de Farey.

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On Eisenstein congruences and beyond

Lecouturier

2023

Notices of the International Consortium of Chinese Mathematicians

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Eisenstein congruences play an important role in modern number theory. We survey some topics related to these congruences, starting from the example of Ramanujan's Delta function modulo 691. This paper does not contain any new results, except Theorem 2.4. This paper is a survey about (higher) Eisenstein congruences and their applications in number theory. As far as the author is aware, the first example of Eisenstein congruences was found by Ramanujan in [46]. Consider the formal series ∆ = q ∏ n≥1 (1 − q n ) 24 = ∑ n≥1 τ(n)q n (for τ(n) ∈ Z). Ramanujan proved that for all prime p we have τ(p) ≡ 1 + p 11 (modulo 691).

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Modular symbols, Eisenstein series, and congruences

Cited by 4 publications

References 14 publications

The Jacquet–Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight $2$

The Jacquet–Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight $2$

Symboles modulaires et produit de Petersson

On Eisenstein congruences and beyond

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