A Shintani-Katok-Sarnak type correspondence for Maass cusp forms of level $N$ is shown to be derived from analytic properties of prehomogeneous zeta functions whose coefficients involve periods of Maass forms.

We consider a natural basis for the space of weakly holomorphic modular forms for $\Gamma_{0}^{+}(3)$. We prove that for some of the basis elements, if $z_{0}$ in the fundamental domain for $\Gamma_{0}^{+}(3)$ is one of zeroes of the elements, then either $z_{0}$ is transcendental or is in $\{\frac{i}{\sqrt{3}}, \frac{-1+\sqrt{2}i}{3}, \frac{-3+\sqrt{3}i}{6}, \frac{-1+\sqrt{11}i}{6}\}$.

In this short note, we study Lehmer’s conjecture in terms of stability theory. We state Bounded Lehmer’s conjecture, and we prove that if a certain formula is uniformly stable in a class of structures, then Bounded Lehmer’s conjecture holds. Our proof is based on Van der Waerden’s theorem from additive combinatorics.