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

Jim Brown; Krzysztof Klosin

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arXiv · 1402.0900

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

Jim Brown·Krzysztof Klosin

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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 $f$.

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BibTeX RIS

Jim Brown, Krzysztof Klosin. 2014-02-04. On the norms of $p$-stabilized elliptic newforms (with an appendix by Keith Conrad). https://arxiv.org/abs/1402.0900

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