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Register a new userHarmonic Maass forms associated to real quadratic fields
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In this paper, we explicitly construct harmonic Maass forms that map to the weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
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Let $q:=e^{2 pi iz}$, where $z in mathbb{H}$. For an even integer $k$, let $f(z):=q^hprod_{m=1}^{infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th Hecke operator and $D$ be a divisor of a modular curve with level $N$. Both subjects, the exponents $c(m)$ of a modular form and the distribution of the points in the support of $T_m. D$, have been widely investigated. When the level $N$ is one, Bruinier, Kohnen, and Ono obtained, in terms of the values of $j$-invariant function, identities between the exponents $c(m)$ of a modular form and the points in the support of $T_m.D$. In this paper, we extend this result to general $Gamma_0(N)$ in terms of values of harmonic weak Maass forms of weight $0$. By the distribution of Hecke points, this applies to obtain an asymptotic behaviour of convolutions of sums of divisors of an integer and sums of exponents of a modular form.
We discuss the problem of the vanishing of Poincare series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujans mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincare series and the connection to Ramanujans mock theta functions.
We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.
We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals.
We give an explicit construct of a harmonic weak Maass form $F_{Theta}$ that is a lift of $Theta^3$, where $Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the holomorphic part of $F_{Theta}$ are associated to class numbers of real quadratic fields.
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