The subject of harmonic weak Maass forms has enjoyed an increasing amount of popularity in recent years. This is partly due to the relationship with mock theta functions and the possibility to interpret these as combinatorial generating functions (cf. e.g. K. Ono [“Unearthing the visions of a master: harmonic Maass forms and number theory”, in: Jerison, David (ed.) et al. Current developments in mathematics, 2008. Somerville, MA: International Press, 347–454 (2009; Zbl 1229.11074)]). It is also known that harmonic weak Maass forms (in the guise of half-integral weight Eisenstein series) can serve as generating functions for class numbers (cf. e.g. F. Hirzebruch and D. Zagier [Invent. Math. 36, 57–113 (1976; Zbl 0332.14009)]).
The theory of harmonic weak Maass forms was first developed (in a systematic way) by J. Bruinier and J. Funke [Duke Math. J. 125, No. 1, 45–90 (2004; Zbl 1088.11030)]. The paper under review is an important step towards a better understanding of this class of functions and its role in the world of automorphic forms and arithmetic geometry.
One of the main results in the current paper is a generalization of the classical result of Waldspurger, Kohnen and Zagier which says that critical values of quadratic twists of a given modular \(L\)-function are determined by the Fourier coefficients of an associated half-integral weight modular form.
It is shown that the derivatives at the critical point of quadratic twists of weight \(2\) modular \(L\)-functions are related to Fourier coefficients of associated half-integral weight harmonic weak Maass forms. The main part of the paper is formulated in the more general setting of vector-valued modular forms for the Weil representation. However, for the sake of clarity, and to avoid the introduction of unnecessary notation, we will state the main results in the simpler setting of scalar-valued modular forms of prime level. I will begin by reviewing some of the classical results. For proofs see e.g. W. Kohnen [Math. Ann. 271, 237–268 (1985; Zbl 0542.10018)].
Let \(G\) be a newform of weight \(2\) and prime level \(p\), that is \(G\in S_{2}^{\mathrm{new}}(\Gamma_{0}\left(p\right))\). From Kohnen’s theory of plus spaces we know that there exists a weight \(\frac{3}{2}\) modular form, \(g\in S_{\frac{3}{2}}^{+}(\Gamma_{0}\left(4p\right))\), which lifts to \(G\) under the Shimura correspondence. Suppose that \(G\) and \(g\) have Fourier coefficients \(b_{G}\left(n\right)\) and \(b_{g}\left(n\right)\). That is to say, that \(G\left(\tau\right)=\sum_{n\geq1}b_{G}\left(n\right)q^{n}\) and \(g\left(\tau\right)=\sum_{n\geq1,n\equiv0,3\left(4\right)}b_{g}\left(n\right)q^{n}\) where \(q=e^{2\pi i\tau}\) for \(\tau\in\mathbb{H}=\left\{ \tau=u+iv\,|\, v>0\right\} \). It is known that the Shimura correspondence commutes with the action of Hecke operators, with the operator \(T_{n}\) acting on \(S_{2}^{\mathrm{new}}\left(p\right)\) corresponding to \(T_{n^{2}}\) acting on \(S_{\frac{3}{2}}^{+}\left(4p\right)\). It is therefore natural to expect that the coefficients \(b_{g}\left(n^{2}\right)\) should be determined by the coefficients of \(G\). Indeed, if \(D<0\) is a fundamental discriminant then \[ b_{g}\left(n^{2}\left|D\right|\right)=b_{g}\left(\left|D\right|\right)\sum_{d|n,\left(d,p\right)=1}\mu\left(d\right)\left(\frac{D}{d}\right)b_{G}\left(\frac{n}{d}\right) \] where \(\mu\) is the Möbius function and \(\left(\frac{D}{\cdot}\right)\) denotes the usual Kronecker symbol. The relationship between the square-free coefficients of \(g\) and the function \(G\) is deeper and much more indirect. Suppose that the \(L\)-function of \(G\), \[ L\left(G,s\right)=\sum b_{G}\left(n\right)n^{-s} \] has a functional equation with sign \(\varepsilon\left(G\right)\), meaning that the completed \(L\)-function \[ \Lambda\left(G,s\right)=(2\pi)^{-s}p^{s/2}\Gamma\left(s\right)L\left(g,s\right) \] satisfies \[ \Lambda\left(G,2-s\right)=\varepsilon\left(G\right)\Lambda\left(G,s\right). \] For a fundamental discriminant \(D\), consider the Dirichlet character \(\chi_{D}=\left(\frac{D}{\cdot}\right)\) and the twisted L-series \[ L\left(G,\chi_{D},s\right)=\sum_{n=1}^{\infty}b_{G}\left(n\right)\chi_{D}\left(n\right)n^{-s}. \] It has a functional equation with sign given by \(\varepsilon\left(G;D\right)=\varepsilon(G)\chi_{D}(p) \text{sgn} (D)\). If \(\varepsilon\left(G;D\right)=1\) then the critical value is given by \[ L\left(G,\chi_{D},1\right)=c\cdot\left|b_{g}\left(|D|\right)\right|^{2} \] where \(c\) is an explicit constant. If \(\varepsilon\left(D\right)=-1\) then it is clear that the critical value vanishes and we are naturally led to study the derivative at the critical point \(L'(G,\chi_{D},1)\).
In order to study this derivative it turns out to be necessary to introduce the space of harmonic weak Maass forms of level \(4p\) and weight \(\frac{1}{2}\). This space, denoted by \(H_{\frac{1}{2}}(4p)\), consists of real analytic functions on the upper half-plane, \(f:\mathbb{H}\rightarrow\mathbb{C}\), satisfying the following conditions:
(a) \(f\left(A\tau\right)=v_{\theta}\left(A\right)\left(c\tau+d\right)^{\frac{1}{2}}f\left(\tau\right)\), for all \(A=\left(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix} \right)\in\Gamma_{0}\left(p\right),\)
(b) \(\Delta_{k}f\left(\tau\right)=0,\) where \(\Delta_{k}=v^{2}\left(\frac{\partial^{2}}{\partial u^{2}}+\frac{\partial^{2}}{\partial v^{2}}\right)-ikv\left(\frac{\partial}{\partial u}+i\frac{\partial}{\partial v}\right)\), and
(c) there exist a polynomial \(P_{f}\in\mathbb{C}\left[q^{-1}\right]\) and \(\epsilon>0\) such that \(f\left(\tau\right)-P_{f}\left(q\right)=O\left(e^{-\varepsilon v}\right)\) as \(v\rightarrow\infty\) (and similar conditions at the other cusps).
Here \(v_{\theta}\left(A\right)=\varepsilon_{d}^{-1}\left(\frac{c}{d}\right)\) is the theta multiplier and \(\varepsilon_{d}=1\) if \(d\equiv1\mod4\) and \(\varepsilon_{d}=i\) otherwise. It can be shown that the differential operator \(\xi_{\frac{1}{2}}\), defined by \(\xi_{\frac{1}{2}}\left(f\right)\left(\tau\right)=2iv^{\frac{1}{2}}\overline{\frac{\partial f}{\partial\overline{\tau}}}\), maps \(H_{\frac{1}{2}}\left(4p\right)\) onto \(S_{\frac{3}{2}}^{+}\left(4p\right)\). We choose an \(f_{g}\in H_{\frac{1}{2}}\left(4p\right)\) such that \(\xi_{\frac{1}{2}}\left(f\right)=\left\| g\right\| ^{-2}g\), where \(\left\| g\right\| \) is the Petersson norm of \(g\). We know that \(f_{g}\) has a Fourier expansion at infinity of the form \[ f_{g}\left(\tau\right)=P_{f}\left(q\right)+\sum_{n>0}c_{g}^{+}\left(n\right)q^{n}+\sum_{n<0}c_{g}^{-}\left(n\right)\Gamma\left(k-1;4\pi\left|n\right|v\right)q^{n}. \]
Using the notation and definitions introduced above we can now formulate the simplest version of the main result of the paper.
Theorem. If \(\Delta\) is a fundamental discriminant with \(\left(\frac{\Delta}{p}\right)=1\) and \(\varepsilon\left(G;\Delta\right)=-1\) then \(L'\left(G,\chi_{\Delta},1\right)=0\) if and only if \(c_{g}^{+}\left(\Delta\right)\) is algebraic.
From the relationship \(\xi_{\frac{1}{2}}\left(f\right)=\left\| g\right\| ^{-2}\cdot g\) it is easy to see that \(b_{g}\left(n\right)=-4\sqrt{\pi n}\left\| g\right\| \cdot c_{g}^{-}\left(-n\right)\) and it follows that the coefficients of the non-holomorphic part are well-understood. In particular, the quotients \(c_{g}^{-}\left(-n\right)/c_{g}^{-}\left(-1\right)\) are all algebraic numbers. In contrast to this situation, the coefficients of the holomorphic part are more mysterious and are believed to be independent transcendental numbers in general. Additionally the coefficients have totally different asymptotic behavior with \(c_{g}^{-}\left(-n\right)\) growing at most polynomially while \(c_{g}^{+}\left(-n\right)=O(e^{c_{1}\sqrt{n}})\), for some constant \(c_{1}>0\), as \(n\to\infty\). For a comprehensive discussion of the algebraicity of harmonic weak Maass forms see K. Ono [Ramanujan J. 20, No. 3, 297–309 (2009; Zbl 1247.11053)].
Although the theorem above can be stated in relatively simple terms, the proof is very intricate and makes heavy use of tools from arithmetic geometry and in particular the theory of Heegner divisors and differentials of the third kind.
The basic ideas behind the key steps can be described as follows: First of all a twisted regularized theta lift is used to lift the harmonic weak Maass form \(f_{g}\) to a Green’s function, \(\Phi_{\Delta,r}\left(z,f_{g}\right)\) corresponding to a certain twisted Heegner divisor. It is then shown that the Fourier coefficients of \(\Phi_{\Delta,r}(z,f_{g})\) (which are directly related to those of \(f_{g}\)) are algebraic precisely if the Heegner divisor \(y_{\Delta,r}(f_{g})\) vanishes in \(J(\mathbb{Q}(\sqrt{\Delta}))\otimes\mathbb{C}\) where \(J\) is the Jacobian of \(\Gamma_{0}\left(p\right)\) (see Theorem 7.6). The final step is to use the Gross-Zagier theorem to establish the equivalence between the vanishing of \(y_{\Delta,r}(f_{g})\) and the vanishing of the derivative \(L'\left(G,\chi_{\Delta},1\right)\) (see Theorem 7.8).
During the course of the proof of the main results the authors also establish many results of independent interest, especially in the theory of generalized twisted theta lifts and Borcherds products (see Sections 5 and 6). In the last section of the paper, the authors provide an extensive collection of examples which illustrate many of the most important results.