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# Periods of Modular Forms and Imaginary Quadratic Base Change

Published:2010-05-11
Printed: Sep 2010
• Mak Trifković,
Mathematics and Statistics, University of Victoria, Victoria, BC, Canada
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## Abstract

Let $f$ be a classical newform of weight $2$ on the upper half-plane $\mathcal H^{(2)}$, $E$ the corresponding strong Weil curve, $K$ a class number one imaginary quadratic field, and $F$ the base change of $f$ to $K$. Under a mild hypothesis on the pair $(f,K)$, we prove that the period ratio $\Omega_E/(\sqrt{|D|}\Omega_F)$ is in $\mathbb Q$. Here $\Omega_F$ is the unique minimal positive period of $F$, and $\Omega_E$ the area of $E(\mathbb C)$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.
 MSC Classifications: 11F67 - Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols