Murmurations in the Depth Aspect for Maass and Modular Forms

Abstract

We study murmurations in the depth aspect for holomorphic cusp forms of conductor ${\ell }^{2a}$ and fixed weight, where $\ell$ is an odd prime. For both ${\mathrm{GL}}_2$ and the definite quaternion algebra ramified at $\{\infty ,\ell \}$, we determine the murmuration density as $a\to \infty$ with $\ell$ fixed. The resulting density agrees with the one previously obtained for odd conductor exponents, and hence gives a uniform density for cusp forms of conductor ${\ell }^n$ as $n\to \infty$. We also consider the case of Maass forms of conductor ${\ell }^n$. Finally, we compute the murmuration density in conductor ${\ell }^n$ as $\ell \to \infty$ with $n\ge 3$ fixed.

42 pages, 3 figures