scholarly journals New Zero-Density Results for Automorphic L-Functions of GL(n)

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2061
Author(s):  
Wenjing Ding ◽  
Huafeng Liu ◽  
Deyu Zhang
Keyword(s):  
Number Field ◽  
Imaginary Part ◽  
Rational Number ◽  
Upper Bound ◽  
Automorphic Representation ◽  
Density Estimates ◽  
Narrow Strip ◽  
Zero Density ◽  
Detecting Method ◽  
Density Results

Let L(s,π) be an automorphic L-function of GL(n), where π is an automorphic representation of group GL(n) over rational number field Q. In this paper, we study the zero-density estimates for L(s,π). Define Nπ(σ,T1,T2) = ♯ {ρ = β + iγ: L(ρ,π) = 0, σ<β<1, T1≤γ≤T2}, where 0≤σ<1 and T1<T2. We first establish an upper bound for Nπ(σ,T,2T) when σ is close to 1. Then we restrict the imaginary part γ into a narrow strip [T,T+Tα] with 0<α≤1 and prove some new zero-density results on Nπ(σ,T,T+Tα) under specific conditions, which improves previous results when σ near 34 and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method.

scholarly journals PRIME DENSITY RESULTS FOR HECKE EIGENVALUES OF A SIEGEL CUSP FORM

2011 ◽  
Vol 07 (04) ◽  
pp. 971-979 ◽  
Author(s):  
ABHISHEK SAHA
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Automorphic Representation ◽  
Hecke Eigenvalues ◽  
Siegel Cusp Form ◽  
Density Results

Let F ∈ Sk( Sp (2g, ℤ)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues μF(n). Suppose that the associated automorphic representation πF is locally tempered everywhere. For each c > 0, we consider the set of primes p for which |μF(p)| ≥ c and we provide an explicit upper bound on the density of this set. In the case g = 2, we also provide an explicit upper bound on the density of the set of primes p for which μF(p) ≥ c.

scholarly journals Gaussian Integers

2013 ◽  
Vol 21 (2) ◽  
pp. 115-125
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Daichi Mizushima ◽  
Yasunari Shidama
Keyword(s):  
Number Field ◽  
Rational Number ◽  
Rational Numbers ◽  
Gaussian Integer ◽  
Algebraic Integers ◽  
Quotient Field ◽  
Gaussian Integers ◽  
Integer Ring

Summary Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic.

scholarly journals Automorphy factors for a Hilbert modular group

1988 ◽  
Vol 30 (2) ◽  
pp. 231-236
Author(s):  
Shigeaki Tsuyumine
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Algebraic Number Field ◽  
Maximal Order ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Totally Real

Let K be a totally real algebraic number field of degree n > 1, and let OK be the maximal order. We denote by гk, the Hilbert modular group SL2(OK) associated with K. On the extent of the weight of an automorphy factor for гK, some restrictions are imposed, not as in the elliptic modular case. Maass [5] showed that the weight is integral for K = ℚ(√5). It was shown by Christian [1] that for any Hilbert modular group it is a rational number with the bounded denominator depending on the group.

scholarly journals On Galois groups of class two extensions over the rational number field

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

scholarly journals On a Certain Function Analogous to log|η(z)|

1970 ◽  
Vol 40 ◽  
pp. 193-211 ◽  
Author(s):  
Tetsuya Asai
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Modular Group ◽  
Classical Case ◽  
Algebraic Number Field ◽  
View Point ◽  
Limit Formula

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

scholarly journals Adjoint L-Functions for GL(3) and U2,1

2019 ◽  
Author(s):  
Joseph Hundley ◽  
Qing Zhang
Keyword(s):  
Number Field ◽  
Automorphic Representation ◽  
Base Change ◽  
Unitary Groups ◽  
Finite Part ◽  
Cuspidal Automorphic Representation ◽  
The Relationship

AbstractWe show that the finite part of the adjoint $L$-function (including contributions from all non-archimedean places, including ramified places) is holomorphic in ${\textrm{Re}}(s) \ge 1/2$ for a cuspidal automorphic representation of ${\textrm{GL}}_3$ over a number field. This improves the main result of [21]. We obtain more general results for twisted adjoint $L$-functions of both ${\textrm{GL}}_3$ and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint $L$-functions, endoscopy, and the structure of the stable base change lifting.

ON A CERTAIN GENERAL EXPONENTIAL SUM

2005 ◽  
Vol 01 (02) ◽  
pp. 183-192 ◽  
Author(s):  
H. MAIER ◽  
A. SANKARANARAYANAN
Keyword(s):  
Rational Number ◽  
Upper Bound ◽  
Exponential Sum ◽  
Multiplicative Functions ◽  
Sum Formula

In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum [Formula: see text] and obtain a non-trivial upper bound when α is a certain type of rational number.

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