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Search: All articles in the CMB digital archive with keyword $L$-functions

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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 2007 (vol 50 pp. 234)

Kuo, Wentang
A Remark on a Modular Analogue of the Sato--Tate Conjecture
The original Sato--Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate a modular analogue of the Sato--Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate--Tate measure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform.

Keywords:$L$-functions, Elliptic curves, Sato--Tate
Categories:11F03, 11F25

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