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1. CMB 2010 (vol 53 pp. 571)
Periods of Modular Forms and Imaginary Quadratic Base Change 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.
Category:11F67 |
2. CMB 2005 (vol 48 pp. 535)
On the Error Term in Duke's Estimate for the Average Special Value of $L$-Functions Let $\FF$ be an orthonormal basis for weight $2$
cusp forms of level $N$. We show that various weighted averages of
special values $L(f \tensor \chi, 1)$ over $f \in \FF$ are equal to $4
\pi c + O(N^{-1 + \epsilon})$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error
term of $O(N^{-1/2}\log N)$.
Categories:11F67, 11F11 |