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Register a new userOn the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups
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Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
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We complete the proof of Proposition 5.3 of [GJR04].
In this paper, we consider the family ${L_j(s)}_{j=1}^{infty}$ of $L$-functions associated to an orthonormal basis ${u_j}_{j=1}^{infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues ${lambda_j=kappa_{j}^{2}+1/4}_{j=1}^{infty}$. We prove the following effective non-vanishing result: At least $50 %$ of the central values $L_j(1/2)$ with $kappa_j leq T$ do not vanish as $Trightarrow infty$. Furthermore, we establish effective non-vanishing results in short intervals.
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime-power conductor.
We look at the values of two Dirichlet $L$-functions at the Riemann zeros (or a horizontal shift of them). Off the critical line we show that for a positive proportion of these points the pairs of values of the two $L$-functions are linearly independent over $mathbb{R}$, which, in particular, means that their arguments are different. On the critical line we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive $c$.
We study simultaneous non-vanishing of $L(tfrac{1}{2},di)$ and $L(tfrac{1}{2},gotimes di)$, when $di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,mathbb{Z})$ and $g$ is a fixed $SL(2,mathbb{Z})$ Hecke cusp form of weight $kequiv 0 pmod 4$.
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