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1. CMB 2017 (vol 60 pp. 329)

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 Online first

Eroǧlu, Münevver Pınar; Argaç, Nurcan
On Identities with Composition of Generalized Derivations
Let $R$ be a prime ring with extended centroid $C$, $Q$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that $dim_{C}(RC) \gt 4$, $f(X_{1},\dots,X_{n})$ a multilinear polynomial over $C$ which is not central-valued on $R$ and $f(R)$ the set of all evaluations of the multilinear polynomial $f\big(X_{1},\dots,X_{n}\big)$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that $G^2\big(u\big)u \in C$ for all $u\in f(R)$ then one of the following conditions holds: (I) there exists $a\in Q$ such that $a^2=0$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$; (II) there exists $a\in Q$ such that $0\neq a^2\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$; (III) $char(R)=2$ and one of the following holds: (i) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$ and $a^{2}=b^{2}\in C$; (ii) there exist $a, b\in Q$ such that $G(x)=ax+xb$ for all $x\in R$, $a^{2}, b^{2}\in C$ and $f(X_{1},\ldots,X_{n})^{2}$ is central-valued on $R$; (iii) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$ and $a^2+d(a)=0$; (iv) there exist $a \in Q$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R$, $d^2=0$, $a^2+d(a)\in C$ and $f(X_{1},\dots,X_{n})^{2}$ is central-valued on $R$. Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying $G^2(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$.

Keywords:prime ring, generalized derivation, composition, extended centroid, multilinear polynomial, maximal right ring of quotients
Categories:16N60, 16N25

3. CMB 2010 (vol 53 pp. 587)

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq
Hulls of Ring Extensions
We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls $\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of $B$ are strongly Morita equivalent as $C^*$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebra
Categories:16N60, 16D90, 16S99, 16S50, 46L05

4. CMB 2005 (vol 48 pp. 561)

Foth, Philip
A Note on Lagrangian Loci of Quotients
We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group.

Keywords:Quotients, involutions, real forms, Lagrangian loci

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