scholarly journals Nonvanishing of L-functions, the Ramanujan Conjecture, and Families of Hecke Characters

2013 ◽  
Vol 65 (1) ◽  
pp. 22-51 ◽  
Author(s):  
Valentin Blomer ◽  
Farrell Brumley
Keyword(s):  
Class Group ◽  
Cusp Forms ◽  
Critical Strip ◽  
Residue Classes ◽  
Ramanujan Conjecture

AbstractWe prove a nonvanishing result for families of GLn× GLn Rankin–Selberg L-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on GLn. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.

On effective determination of symmetric-square lifts

2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Qingfeng Sun
Keyword(s):  
Cusp Forms ◽  
Maass Forms ◽  
Central Values ◽  
Laplace Eigenvalue ◽  
Symmetric Square ◽  
Ramanujan Conjecture

AbstractLet F be the symmetric-square lift with Laplace eigenvalue λ F (Δ) = 1+4µ2. Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ⋇ h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ≍ϱ+ɛ, t9 = max {4(1+4θ)/(1−18θ), 8(2−9θ)/3(1−18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL2 Maass forms.

Nonvanishing of derivatives of Hecke $$L$$L-functions associated to cusp forms inside the critical strip

2018 ◽  
Vol 51 (2) ◽  
pp. 319-327
Author(s):  
Winfried Kohnen ◽  
Jyoti Sengupta ◽  
Miriam Weigel
Keyword(s):  
Cusp Forms ◽  
Critical Strip ◽  
Derivatives Of

scholarly journals Stably free resolutions of lattices over finite groups

1990 ◽  
Vol 49 (3) ◽  
pp. 364-385
Author(s):  
K. W. Gruenberg
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Class Group ◽  
Free Resolution ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Residue Classes ◽  
Projective Class

AbstractFor a ZG-lattice A, the nth partial free Euler characteristic εn(A) is defined as the infimum of all where F* varies over all free resolutions of A. It is shown that there exists a stably free resolution E* of A which realises εn(A) for all n≥0 and that the function n → εn(A) is ultimately polynomial no residue classes. The existence of E* is established with the help of new invariants σn(A) of A. These are elements in certain image groups of the projective class group of ZG. When ZG allows cancellation, E* is a minimal free resolution and is essentially unique. When A is periodic, E* is ultimately periodic of period a multiple of the projective period of A.

scholarly journals BEYOND ENDOSCOPY VIA THE TRACE FORMULA – III THE STANDARD REPRESENTATION

2018 ◽  
Vol 19 (4) ◽  
pp. 1349-1387 ◽  
Author(s):  
S. Ali Altuğ
Keyword(s):  
Number Theory ◽  
Asymptotic Expansion ◽  
Trace Formula ◽  
Automorphic Forms ◽  
Fourier Transforms ◽  
Cusp Forms ◽  
Selberg Trace Formula ◽  
Johns Hopkins University ◽  
Ramanujan Conjecture ◽  
Beyond Endoscopy

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

ON THE DIVISOR CLASS GROUP OF 3-FOLD SINGULARITIES

2003 ◽  
Vol 14 (01) ◽  
pp. 105-117 ◽  
Author(s):  
MIREL CAIBĂR
Keyword(s):  
Class Number ◽  
Class Group ◽  
Newton Polyhedron ◽  
Divisor Class Group ◽  
Divisor Class ◽  
Canonical Singularity ◽  
Residue Classes

In this note we calculate the divisor class number of an isolated canonical singularity [Formula: see text], which is assumed to be nondegenerate with respect to its Newton polyhedron, in terms of a suitable set of monomials whose residue classes form a basis for the Milnor algebra of f.

scholarly journals Some Riemann Hypotheses from random walks over primes

2018 ◽  
Vol 20 (07) ◽  
pp. 1750085 ◽  
Author(s):  
Guilherme França ◽  
André LeClair
Keyword(s):  
Functional Equation ◽  
Critical Line ◽  
Arithmetic Function ◽  
Prime Numbers ◽  
Cusp Forms ◽  
Euler Product ◽  
Critical Strip ◽  
One Dimensional ◽  
The Right ◽  
Multiplicative Characters

The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes, we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is [Formula: see text], then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this [Formula: see text] growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results, we obtain an equation which relates every individual non-trivial zero of the [Formula: see text]-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet [Formula: see text]-functions due to the existence of the pole at [Formula: see text], in which the Riemann [Formula: see text]-function is a particular case.

scholarly journals THE SIGN OF FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT CUSP FORMS

2012 ◽  
Vol 08 (03) ◽  
pp. 749-762 ◽  
Author(s):  
THOMAS A. HULSE ◽  
E. MEHMET KIRAL ◽  
CHAN IEONG KUAN ◽  
LI-MEI LIM
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Square Root ◽  
Critical Strip ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Finite Set ◽  
Invent Math

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175–198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 𝔣 are, up to a finite set of factors, one of [Formula: see text] when m is square-free and f is the integral weight cusp form related to 𝔣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440–481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

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