Simultaneous nonvanishing of automorphic L-functions at the central point

Let g be a holomorphic Hecke eigenform and {u _j } an orthonormal basis of even Hecke?CMaass forms for ${\textup{SL}(2,\mathbb{Z})}$ . Denote L(s, g × u _j ) and L(s, u _j ) the corresponding L-functions. In this paper, we give an asymptotic formula for the average of ${L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)}$ , from which we derive that there are infinitely many u _j ??s such that ${L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)\neq0}$ .     

Twisted moments of automorphic L-functions

We study the moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the square of the value of the standard L-function at the center of the critical strip. We give a precise expansion of the moments as the order goes to infinity.     

On the special values of automorphic L-functions

Let π be a cuspidal representation of $\text{GL}{}_{\mathrm{n}}\text{(}\text{A}{}_{\mathrm{<mtext>Q</mtext>}}\text{)}$ with non-vanishing cohomology and denote by L(π,s) its L-function. Under a certain local non-vanishing assumption, we prove the rationality of the values of L(π?χ,0) for characters χ, which are critical for π. Note that conjecturally any motivic L-function should coincide with an automorphic L-function on GL_n; hence, our result corresponds to a conjecture of Deligne for motivic L-functions. To cite this article: J. Mahnkopf, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

On the Selberg orthogonality for automorphic L-functions

For automorphic L-functions L(s, π) and L( s,p^￠){L( s,\pi^{\prime })} attached to automorphic irreducible cuspidal representations π and π′ of GL_m( \mathbbQ_A){GL_{m}( \mathbb{Q}_{A})} and GL_m^￠(\mathbbQ_A) {GL_{m^{\prime }}(\mathbb{Q}_{A}) }, we prove the Selberg orthogonality unconditionally for m ≤ 4 and m′ ≤ 4, and under hypothesis H of Rudnik and Sarnak if m > 4 or m′ > 4, without the additional requirement that at least one of these representations be self-contragradient.     

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factoriz...     

A proof of Selberg's orthogonality for automorphic L-functions

Let π and π′ be automorphic irreducible cuspidal representations of GL_m(Q_A) and GL_m′(Q_A), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].     

Moments of the products of quadratic twists of automorphic L-functions

In this paper, assuming the Generalized Riemann Hypothesis and some other hypotheses, we give sharp upper bounds for the moments of the products of central values of automorphic L-functions twisted by quadratic characters and averaged over fundamental discriminants.     

Mollification of the fourth moment of automorphic L-functions and arithmetic applications

We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular those obtained by base change from GL ₂(Q) to GL ₂(K) for K a cyclic field of low degree. Oblatum 22-VI-1999 & 3-III-2000?Published online: 5 June 2000     

Values of twisted Artin L-functions

This note gives a simple proof that certain values of Artin’s L-function, for a representation ρ with character χ _ρ , are stable under twisting by an even Dirichlet character χ, up to the dim(ρ)th power of the Gauss sum τ(χ) and an element generated over \({\mathbb{Q}}\) by the values of χ and χ _ρ . This extends a result due to J. Coates and S. Lichtenbaum.     

Periods of automorphic forms, poles of L-functions and functorial lifting

In this paper, we survey our work on period, poles or special values of certain automorphic L-functions and their relations to certain types of Langlands functorial transfers. We outline proofs for some results and leave others as conjectures.     

Values of binary linear forms at prime arguments

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence V and δ>0, let E(V, X, δ) denote the number of υ∈V with υ≤X for which the inequality |λ1p1+λ2ρ2−υ|<υ−δ has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E(V, X, δ) «max(X35+2δ+ε,X23+43δ+ε).     

On the zeros on the critical line of L-functions corresponding to automorphic cusp forms

We consider an automorphic cusp form of integer weight k ≥ 1, which is the eigenfunction of all Hecke operators. It is proved that, for the L-series whose coefficients correspond to the Fourier coefficients of such an automorphic form, the positive fraction of nontrivial zeros lie on the critical line.     

Zero density estimates for automorphic L-functions of {GL_{m}}

We study the zero-density estimates for automorphic L-functions \({L(s, \pi)}\) for GL _m when \({\sigma}\) is near 1. In particular, we get a range of \({\sigma}\) for which the density hypothesis holds. The proofs use a zero detecting argument, the Halász–Montgomery inequality and a bound for an integral power moment of \({L(1/2+it, \pi)}\).     

实二次数域的一类L-函数在中心点的值

为了证明L（1/2，x）≠0，本文定义了实二次域的一类L-函数并给出其在在中心点≥处的值的表达式．     

The distribution of the zeros of L-functions and the least prime in some arithmetic progression

This paper gives some new estimations to the distribution of the zeros of L-functions and proves that the least prime in an arithmetic progression with a prime differenceq is ≪q ^4.5.