An Elementary Proof of a Weak Exceptional Zero Conjecture

2004 ◽  
Vol 56 (2) ◽  
pp. 373-405 ◽  
Author(s):  
Louisa Orton
Keyword(s):  
Elementary Proof ◽  
Central Point ◽  
Cusp Forms ◽  
Dirichlet Characters

AbstractIn this paper we extend Darmon's theory of “integration on ℋp × ℋ” to cusp forms f of higher even weight. This enables us to prove a “weak exceptional zero conjecture”: that when the p-adic L-function of f has an exceptional zero at the central point, the ℒ-invariant arising is independent of a twist by certain Dirichlet characters.

scholarly journals Subconvexity for a double Dirichlet series and non-vanishing of L-functions

2018 ◽  
Vol 14 (06) ◽  
pp. 1573-1604
Author(s):  
Alexander Dahl
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Dirichlet Series ◽  
Upper Bound ◽  
Dihedral Group ◽  
Central Point ◽  
Double Dirichlet Series ◽  
Dirichlet Characters ◽  
Equation Group

We study a double Dirichlet series of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are quadratic Dirichlet characters with prime conductors [Formula: see text] and [Formula: see text] respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to [Formula: see text]. The developed theory is used to prove an upper bound for the smallest positive integer [Formula: see text] such that [Formula: see text] does not vanish. Additionally, a convexity bound at the central point is established to be [Formula: see text] and a subconvexity bound of [Formula: see text] is proven. An application of bounds at the central point to the non-vanishing problem is also discussed.

AN ELEMENTARY PROOF OF THE CONDUCTOR-DISCRIMINANT FORMULA

2010 ◽  
Vol 06 (05) ◽  
pp. 1191-1197
Author(s):  
GABRIEL VILLA-SALVADOR
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Prime Number ◽  
Elementary Proof ◽  
Abelian Extension ◽  
Roots Of Unity ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Dirichlet Characters ◽  
Ramification Index

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pn-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

BOUNDS FOR A MEAN VALUE OF CHARACTER SUMS

2008 ◽  
Vol 04 (02) ◽  
pp. 249-293 ◽  
Author(s):  
NIGEL WATT
Keyword(s):  
Prime Factor ◽  
Upper Bounds ◽  
Mean Value ◽  
Cusp Forms ◽  
Smooth Functions ◽  
Mean Values ◽  
Carmichael Numbers ◽  
Dirichlet Characters ◽  
Dirichlet Polynomials ◽  
The Mean

We obtain new upper bounds for the mean squared modulus of sums ∑n∈ℕAnχ(n), where the sequence (An) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (cm) satisfying cm = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has An = ∑uvm = n α(u)β(v)cm (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

scholarly journals MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

2021 ◽  
pp. 1-45
Author(s):  
RIZWANUR KHAN ◽  
MATTHEW P. YOUNG
Keyword(s):  
Spectral Parameter ◽  
Critical Line ◽  
Short Interval ◽  
Central Point ◽  
Cusp Forms ◽  
Sharp Bounds ◽  
Second Moment ◽  
Symmetric Square

Abstract We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

AN ELEMENTARY PROOF OF A THEOREM OF LEE

1991 ◽  
Vol 11 (3) ◽  
pp. 356-360 ◽  
Author(s):  
Jia'an Yan
Keyword(s):  
Elementary Proof

Congruences between Hilbert Cusp Forms

2018 ◽  
Author(s):  
Hiroshi Saito ◽  
Masatoshi Yamauchi
Keyword(s):  
Cusp Forms
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