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3 precised question

By a Maass form I just mean--maybe a bit loosely--any real analytic $\Bbb C$-valued function $f$ on the upper halfplane $\cal{H}$ which is automorphic of weight $k\in\Bbb Z$ with respect to a discrete subgroup $\Gamma<{\rm SL}_2(\Bbb R)$ such that $\Gamma\backslash\cal H$ has finite volume, and an eigenfunction for the Laplacian operator corresponding to the Casimir element in the universal enveloping algebra of the complexified $\rm{sl}_2$.

Is it true that the zeroes of these forms are isolated?

The answer is obviously affirmative in the case of holomorphic modular forms.

$\textbf{Edit}$: Scott's comment and Matt's answer below show that the answer is generally negative when the weight is $0$. Then, one can construct real valued Maass forms which have nodal curves.

Thus, let's make the assumption that $k\neq0$ and in particular that the Maass form $f$ belongs to a discrete series representation space (generated by a holomorphic modular form).

To make things as explicit as possible assume also that $\Gamma$ is either a congruence subgroup of ${\rm SL}_2(\Bbb Z)$ or the group of norm 1 elements in an Eichler order of an indefinite quaternion algebra over $\Bbb Q$

2 eliminated ambiguity: the Casimir element is not an element of the Lie algebra

By a Maass form I just mean--maybe a bit loosely--any real analytic $\Bbb C$-valued function $f$ on the upper halfplane $\cal{H}$ which is automorphic of weight $k\in\Bbb Z$ with respect to a discrete subgroup $\Gamma<{\rm SL}_2(\Bbb R)$ such that $\Gamma\backslash\cal H$ has finite volume, and an eigenfunction for the Laplacian operator corresponding to the Casimir element in the Lie universal enveloping algebra of the complexified $\rm{SL}_2$.\rm{sl}_2$. Is it true that the zeroes of these forms are isolated? The answer is obviously affirmative in the case of holomorphic modular forms. 1 # Zeroes of Maass forms By a Maass form I just mean--maybe a bit loosely--any real analytic$\Bbb C$-valued function$f$on the upper halfplane$\cal{H}$which is automorphic of weight$k\in\Bbb Z$with respect to a discrete subgroup$\Gamma<{\rm SL}_2(\Bbb R)$such that$\Gamma\backslash\cal H$has finite volume, and an eigenfunction for the Laplacian operator corresponding to the Casimir element in the Lie algebra of$\rm{SL}_2\$.

Is it true that the zeroes of these forms are isolated?

The answer is obviously affirmative in the case of holomorphic modular forms.