Integral traces of singular values of weak Maass forms

Duke, William; Jenkins, Paul A.

doi:10.2140/ant.2008.2.573

Cited by 42 publications

(58 citation statements)

References 19 publications

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“…Remark. Note that these results generalize works of Duke and Jenkins [DJ08] and Miller and Pixton [MP10]. We also obtain the results in Section 9 of [Zag02] as a special case.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 83%

“…Zagier's results were generalized in various directions, mostly for modular curves of genus 0 [BO07,DJ08,Kim09,MP10]. Building upon previous work of Funke [Fun02] Bruinier and Funke [BF06] showed that Zagier's result on the traces of the j-function can be obtained as a special case of a theta lift using a kernel function constructed by Kudla and Millson [KM86].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Formulas for the coefficients of half-integral weight harmonic Maaß forms

Alfes

2014

Math. Z.

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Abstract. Recently, Bruinier and Ono proved that the coefficients of certain weight −1/2 harmonic weak Maaß forms are given as "traces" of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight 3/2 + k, k even, and weight 1/2 − k, k odd, by extending the theta lift of Bruinier-Funke and Bruinier-Ono. Moreover, we generalize these results to include twisted traces of singular moduli using earlier work of the author and Ehlen on the twisted Bruinier-Funke-lift. Employing a duality result between weight k and 2 − k, we are able to cover all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight 1/2 − k, k odd, is connected to the vanishing of the special value of the L-function of a certain derivative of the lifted function.

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“…Remark. Note that these results generalize works of Duke and Jenkins [DJ08] and Miller and Pixton [MP10]. We also obtain the results in Section 9 of [Zag02] as a special case.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 83%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Formulas for the coefficients of half-integral weight harmonic Maaß forms

Alfes

2014

Math. Z.

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“…We apply this to F (z) which is a weight −2 weakly holomorphic modular form, a meromorphic modular form whose poles are supported at cusps. Theorem 3.6 is a new result which adds to the extensive literature (for example, see [5,9,10,13,14,15,22,29,30]) inspired by Zagier's seminal paper [34] on "traces" of singular moduli.…”

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confidence: 99%

Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

Bruinier

Ono

2013

Advances in Mathematics

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Abstract. We prove that the coefficients of certain weight −1/2 harmonic Maass forms are "traces" of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight −2 harmonic weak Maass forms to spaces of weight −1/2 vectorvalued harmonic weak Maass forms on Mp 2 (Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant −24n + 1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.

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“…(3) As pointed out by the referee, in the absence of holomorphic cusp forms (k < 6), an obvious modification of the theorem (with f = 0 and arbitrary λ p ) remains true for k ≥ 2. This follows from (3.4) and was essentially proven by Duke and Jenkins [8].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 53%

“…However, the authors [4] found an appropriate definition of J κ ⊆ S ! κ which makes use of so-called Zagier lifts [8,15], whose main properties where discovered by Duke and Jenkins [8].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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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Integral traces of singular values of weak Maass forms

Cited by 42 publications

References 19 publications

Formulas for the coefficients of half-integral weight harmonic Maaß forms

Formulas for the coefficients of half-integral weight harmonic Maaß forms

Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

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

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