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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 Online first

Le Fourn, Samuel
Nonvanishing of central values of $L$-functions of newforms in $S_2 (\Gamma_0 (dp^2))$ twisted by quadratic characters
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $\chi$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(\Gamma_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes \chi,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes \chi, \cdot)$ and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Keywords:nonvanishing of $L$-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians
Categories:14J15, 11F67

2. 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.

Category:11F67

3. 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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