2013
DOI: 10.1016/j.aim.2013.05.028
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Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

Abstract: 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, w… Show more

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Cited by 76 publications
(93 citation statements)
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References 27 publications
(32 reference statements)
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“…Adding more bosons would not increase the number of degenerate states, since it is impossible to introduce new states with the same energy as the previous ones. This is due to the finite ways of decomposing N E as a sum of positive integers, without considering the order, that is, the number of partitions p(N E ) [31,32].…”
Section: Unperturbed Energy Statesmentioning
confidence: 99%
“…Adding more bosons would not increase the number of degenerate states, since it is impossible to introduce new states with the same energy as the previous ones. This is due to the finite ways of decomposing N E as a sum of positive integers, without considering the order, that is, the number of partitions p(N E ) [31,32].…”
Section: Unperturbed Energy Statesmentioning
confidence: 99%
“…For example, Funke and the first author [11] used the Kudla-Millson theta lift from weight 0 to weight 3/2 harmonic Maass forms to give a new proof and generalizations of Zagier's [23] famous result on the modularity of the generating series of traces of singular moduli. Further, in [13], Ono and the first author used a variant of the Kudla-Millson theta lift to find a finite algebraic formula for the partition function p(n) in terms of traces of CM-values of a certain non-holomorphic modular function. Recently, a similar theta lift was used in [4] to prove a refinement of a theorem of [12] connecting the vanishing of the central derivative of the twisted L-function of an even weight newform and the rationality of some coefficient of the holomorphic part of a half-integral weight harmonic Maass form.…”
Section: Introductionmentioning
confidence: 99%
“…They found mock modular forms of weight 1/2 that are not classical mock theta functions but whose coefficients contain information about the L-series of the curves. Related to this are interesting questions about the algebraicity of the coefficients of harmonic Maass forms (see [29], [8], and their references).…”
Section: Harmonic Maass Formsmentioning
confidence: 99%