scholarly journals Nonvanishing of symmetric square $L$-functions of cusp forms inside the critical strip

1999 ◽  
Vol 128 (6) ◽  
pp. 1641-1646 ◽  
Author(s):  
Winfried Kohnen ◽  
Jyoti Sengupta
Keyword(s):  
Cusp Forms ◽  
Critical Strip ◽  
Symmetric Square

scholarly journals Symmetric square L-values and dihedral congruences for cusp forms

2010 ◽  
Vol 130 (9) ◽  
pp. 2078-2091 ◽  
Author(s):  
Neil Dummigan ◽  
Bernhard Heim
Keyword(s):  
Cusp Forms ◽  
Symmetric Square

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 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.

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.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

2009 ◽  
Vol 05 (07) ◽  
pp. 1321-1345 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Modular Forms ◽  
Prime Order ◽  
Galois Representations ◽  
Critical Value ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Fourier Expansions ◽  
Symmetric Square ◽  
Vector Valued ◽  
Siegel Cusp Form

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

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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