IJPAM: Volume 68, No. 3 (2011)

MOCK HECKE EIGENFUNCTIONS AND
A CONJECTURE OF HEJHAL

C.J. Mozzochi
P.O. Box 1424, Princeton, NJ 08542, USA
e-mail: [email protected]


Abstract. Let $\Gamma=\SL(2,Z)$. Hejhal has formulated a conjecture for Maass waveforms $\Psi(z,R)$ on $\Gamma\backslash H$, namely, that the $\Psi(z,R)$ ``go Gaussian'' as $R\to\infty$, where $R=\sqrt{\lambda-\frac14\,}$.

Let $f(z,k)$ be a Hecke eigenfunction of even weight $k\Ge 2$ on $\Gamma\backslash H$.

\begin{displaymath}
f(z,k)=\sum\limits_{n=1}^\infty
\lambda_n n^{\frac{k-1}{2}}e(nz).
\end{displaymath}

Let $\Psi(z,k)=y^{k/2}\vert f(z,k)\vert$.

In this paper we first replace the $\lambda_n$ with an arbitrary family of independent random variables $\lambda_n(\omega)$ on an arbitrary probability space and show that within our probability model, if the $\Psi(z,k)$ ``go Gaussian'' as $k\to\infty$, they do not do so uniformly.

We then briefly discuss a specific model based on the Sato-Tate probability measure.

Received: November 2, 2010

AMS Subject Classification: 11F03, 11F11, 11F12, 11F72

Key Words and Phrases: modular group, pair correlation, eigenvalues, Laplacian, Selberg trace formula, Bruggeman-Kuznetsov trace formula

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 68
Issue: 3