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            -adic coupling of mock modular forms and shadows

Guerzhoy, Pavel; Kent, Zachary A.; Ono, Ken

doi:10.1073/pnas.1001355107

Cited by 28 publications

(62 citation statements)

References 9 publications

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“…Here we illustrate Theorem 2 for the prime p = 5 and the newform with conductor 27. Let We prove this theorem using techniques outlined in [5]. Similar results can be found in both [1,6].…”

Section: Theoremsupporting

confidence: 51%

Modular Forms and Weierstrass Mock Modular Forms

Clemm

2016

Mathematics

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Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves "encode" the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin's U-operator.

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“…Here we illustrate Theorem 2 for the prime p = 5 and the newform with conductor 27. Let We prove this theorem using techniques outlined in [5]. Similar results can be found in both [1,6].…”

Section: Theoremsupporting

confidence: 51%

Modular Forms and Weierstrass Mock Modular Forms

Clemm

2016

Mathematics

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“…Roughly speaking, we show that the p 2n mth coefficient of a weakly holomorphic Hecke eigenform with algebraic coefficients is congruent to the mth coefficient of a holomorphic Hecke eigenform modulo a high power of p. A similar relation for integral weight modular forms was recently proven in [11] and was later shown by the authors [3] to be related to the occurrence of a p-adic modular form. The similarity between the integral and halfintegral weight cases is far from being obvious.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 79%

Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms

2015

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In this paper, we consider p-adic limits of β −n g|U n p 2 for half-integral weight weakly holomorphic Hecke eigenforms g with eigenvalue λ p = β + β under T p 2 and prove that these equal classical Hecke eigenforms of the same weight. This result parallels the integral weight case, but requires a much more careful investigation due to a more complicated structure of half-integral weight weakly holomorphic Hecke eigenforms.

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“…The argument below is a refinement of similar arguments from the proof of Theorem 1.1 in [4] and the proof of Theorem 1.2 in [3] adapted for our current purposes.…”

Section: The Function M(τ ) Has At Most Linear Exponential Growth At mentioning

confidence: 99%

“…The mock modular form M + has a Fourier expansion in q = e 2πiτ , and the coefficients of this expansion are the subject of interest in this paper. Theorem 1.1 of [3] guaranties the existence of α ∈ R such that…”

Section: The Function M(τ ) Has At Most Linear Exponential Growth At mentioning

confidence: 99%

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On Zagier’s adele

Guerzhoy

2014

Res Math Sci

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Purpose: Don Zagier suggested a natural construction, which associates a real number and p-adic numbers for all primes p to the cusp form g = of weight 12. He claimed that these quantities constitute a rational adele. In this paper we prove this statement, and, more importantly, a similar statement when g is a weight 2 primitive form with rational integer Fourier coefficients. Methods: While a simple modular argument suffices for the proof of Zagier's original claim, consideration of the case when g is of weight 2 involves Hodge decomposition for the formal group law of the rational elliptic curve associated with g. Results and Conclusions: While in the weight 12 setting considered by Zagier the claim under consideration depends on a specific choice of a mock modular form which is good for g, in the case when g is of weight 2, the statement has a global nature, and depends on the fact that the classical addition law for the Weierstrass ζ -function is defined over Z[ 1/6].

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p -adic coupling of mock modular forms and shadows

Cited by 28 publications

References 9 publications

Modular Forms and Weierstrass Mock Modular Forms

Modular Forms and Weierstrass Mock Modular Forms

Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms

On Zagier’s adele

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