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Register a new userSchneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
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Let $j(z)$ be the modular $j$-invariant function. Let $tau$ be an algebraic number in the complex upper half plane $mathbb{H}$. It was proved by Schneider and Siegel that if $tau$ is not a CM point, i.e., $[mathbb{Q}(tau):mathbb{Q}] eq2$, then $j(tau)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $Gamma_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $tau$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i infty$. We prove, under a mild assumption on $f$, that for any fixed $tau$, if $N$ is a prime such that $ Ngeq 23 text{ and } N ot in {23, 29, 31, 41, 47, 59, 71},$ then $f(T_m.tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.
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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.
In this paper, considering the Eichler-Shimura cohomology theory for Jacobi forms, we study connections between harmonic Maass-Jacobi forms and Jacobi integrals. As an application we study a pairing between two Jacobi integrals, which is defined by special values of partial $L$-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as analogue of Haberland formula of elliptic modular forms for Jacobi forms.
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