On the Petersson norm $⟨{\theta }_{\psi },{\theta }_{\psi }⟩$

Abstract

Let $K$ be an imaginary quadratic field of discriminant~$-d<-11$, and for $\ell \ge 1$, let $\psi$ be a Hecke character of $K$ with trivial finite conductor and infinite type~$(2\ell ,0)$. In this work we prove that for any prime $p>3$, the quotient $$\frac{⟨{\theta }_{\psi },{\theta }_{\psi }⟩}{{\Omega }_K^{4\ell }},$$ is $\lambda$-integral for a prime ideal $\lambda$ over $p$, where $⟨{\theta }_{\psi },{\theta }_{\psi }⟩$ is the Petersson norm of the theta function ${\theta }_{\psi }={\theta }_{\psi }(\tau )$ attached to $\psi$, and ${\Omega }_K$ denotes the Chowla--Selberg period attached to~$K$, and the product $$\prod _{i=1}^{h_K}\frac{⟨{\theta }_{{\psi }_i},{\theta }_{{\psi }_i}⟩}{{\Omega }_K^{4\ell }}$$ is rational, where the product is over all $h_K$ of the underlying Hecke characters.