Algebraicity of harmonic Maass forms
This excellent expository article describes recent work by the author, J. H. Bruinier and R. Rhoades [\textit{J. H. Bruinier, K. Ono} and \textit{R. C. Rhoades}, Math. Ann. 342, No. 3, 673--693 (2008); corrigendum 345, No. 1, 31 (2009; Zbl 1246.11098); \textit{J. Bruinier} and \textit{K. Ono}, Ann. Math. (2) 172, No. 3, 2135--2181 (2010; Zbl 1244.11046)] on algebraicity of coefficients of certain harmonic weak Maaß forms and the relation of their results to the following two classical topics: {1.} vanishing of Hecke eigenvalues of integer weight modular forms, {2.} vanishing of derivatives of modular \(L\)-functions. As an example, the first result sheds new light on Lehmer's conjecture on the non-vanishing of Ramanujan's \(\tau\)-function. Let \(\Delta(z)\) be the modular discriminant, that is, the unique cusp form of weight 12 for \(\mathrm{SL}_2(\mathbb{Z})\). Then, \[ \Delta(z) = \sum_{n=1}^\infty \tau(n) q^n, \] where \(z \in \mathbb{H} = \{z\in \mathbb C: \text{Im}(z) > 0 \}\), the complex upper half-plane, and \(q=e^{2\pi iz}\). Theorem 1.2 of the article under review implies that if \(\tau(p)=0\) for a prime \(p\), then, for all \(n\) such that \(\mathrm{ord}_p(n)\) is odd, the \(n\)-th Fourier coefficient of a certain weakly holomorphic Poincaré series is rational. In fact, this is equivalent to the rationality of the first Fourier coefficient. The author provides numerical evidence supporting the conjecture that this coefficient is not rational. The second topic is covered by the results of the author and J. H. Bruinier [Zbl 1244.11046] which relates the vanishing of the quadratic twist of discriminant \(D\) of a Hecke \(L\)-function for a weight 2 newform to the algebraicity of the coefficient of index \(D\) of the holomorphic part of a certain harmonic weak Maaß\ form. In fact, the authors show that harmonic weak Maaß\ forms serve as generating functions for central values and derivatives of quadratic twists of \(L\)-functions of weight \(2\) newforms. This generalizes works of Waldspurger, Kohnen and Zagier [\textit{J.-L. Waldspurger}, J. Math. Pures Appl. (9) 60, 375--484 (1981; Zbl 0431.10015), \textit{W. Kohnen}, Math. Ann. 271, 237--268 (1985; Zbl 0542.10018), \textit{W. Kohnen} and \textit{D. Zagier}, Invent. Math. 64, 175--198 (1981; Zbl 0468.10015)]. Harmonic weak Maaß\ forms were initially introduced by Bruinier and Funke in their work on theta liftings [\textit{J. H. Bruinier} and \textit{J. Funke}, Duke Math. J. 125, No. 1, 45--90 (2004; Zbl 1088.11030)]. Now they play an important role in many different subjects. For instance, they appear in the study of Ramanujan's mock theta functions, and also in arithmetic geometry, combinatorics, physics, and more. We also refer to the paper (and its extensive review in Zbl 1244.11046) by Bruinier and Ono.
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