On the norms of $p$-stabilized elliptic newforms (with an appendix by Keith Conrad)

TL;DR

This paper computes the norms of p-stabilized elliptic newforms and their U_p operators, providing a local factorization of the Petersson norm using classical and adelic methods, and analyzing related L-functions.

Contribution

It introduces explicit calculations of norms for p-stabilized forms and U_p operators, and derives a local factorization of the Petersson norm using these results.

Findings

Calculated the norm of p-stabilized forms both classically and adelically.

Derived a local factorization of the Petersson norm of elliptic newforms.

Connected norm calculations with properties of the symmetric square L-function.

Abstract

Let $f \in S_{\kappa}(\Gamma_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $\lambda_f(p)$ for a prime $p$ not dividing $N$. Let $\alpha_p$ and $\beta_p$ be complex numbers satisfying $\alpha_p + \beta_p = \lambda_f(p)$ and $\alpha_p \beta_p = p^{\kappa-1}$. We calculate the norm of $f_{p}^{\alpha_p}(z) = f(z) - \beta_{p} f(pz)$ as well as the norm of $U_p f$, both classically and adelically. We use these results along with some convergence properties of the Euler product defining the symmetric square L-function of $f$ to give a `local' factorization of the Petersson norm of $f$.