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Search: MSC category 11F67 ( Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols )

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1. CMB 2010 (vol 53 pp. 571)

Trifković, Mak
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.


2. CMB 2005 (vol 48 pp. 535)

Ellenberg, Jordan S.
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

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