Arithmetic Traces of Non-Holomorphic Modular Invariants

Miller, Alison Beth; Pixton, Aaron

doi:10.1142/s1793042110002818

Cited by 31 publications

(42 citation statements)

References 6 publications

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“…After this paper was written, we learned that results similar to some of those presented here were obtained independently in [Miller and Pixton 2007]. We thank the referee for some helpful comments.…”

Section: Acknowledgmentssupporting

confidence: 66%

Integral traces of singular values of weak Maass forms

Duke

Jenkins

2008

ANT

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We define traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of half-integral weight. If the coefficients of f are integral, then these traces are integral as well. We obtain a negative weight analogue of the classical Shintani lift and give an application to a generalization of the Shimura lift.

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Section: Acknowledgmentssupporting

confidence: 66%

Integral traces of singular values of weak Maass forms

Duke

Jenkins

2008

ANT

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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%

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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“…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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Arithmetic Traces of Non-Holomorphic Modular Invariants

Cited by 31 publications

References 6 publications

Integral traces of singular values of weak Maass forms

Integral traces of singular values of weak 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

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