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تسجيل مستخدم جديدSchneider-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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