Non - vanishing of the Central Derivative of Canonical Hecke L - functions ( Math
نویسندگان
چکیده
Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them). Let L(s, χ) denote the Hecke L-function of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a canonical Hecke character with W (χ) = 1, then the central value Λ(1, χ) 6= 0 by a theorem of Montgomery and Rohrlich [MR]. Of course, it automatically vanishes when W (χ) = −1 by the functional equation. The main result of this paper is
منابع مشابه
Non-vanishing of the Central Derivative of Canonical Hecke L-functions
Every Hecke character of K satisfying (1.1) and (1.2) is actually a quadratic twist of a canonical Hecke character (see Section 2 for a precise description of these characters and which fields have them). Let L(s, χ) denote the Hecke L-function of χ, and Λ(s, χ) its completion; Λ(s, χ) satisfies the functional equation Λ(s, χ) = W (χ)Λ(2 − s, χ), where W (χ) = ±1 is the root number. If χ is a c...
متن کاملVanishing and non-vanishing Dirichlet twists of L-functions of elliptic curves
Let L(E/Q, s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E, 1, χ) of the twisted L-function as χ ranges over Dirichlet characters of given order.
متن کاملOn Hecke L-functions attached to half-integral weight modular forms
We would like to recall that in the case of Hecke eigenforms on Γ1 non-vanishing results for their Hecke L-functions at an arbitrary point s0 in the critical strip (not on the critical line) have been proved in [4] (cf. also [7]), using holomorphic kernel functions. This method was carried over to the case of half-integral weight in [8], for arbitrary level. However, in this approach for given ...
متن کاملQuantitative Nonvanishing of L-series Associated to Canonical Hecke Characters
We prove quantitative nonvanishing theorems for central values and central derivatives of L–series associated to canonical Hecke characters of imaginary quadratic fields. These results have applications to the study of Chow groups of Kuga-Sato varieties. Some key ingredients in the proofs are bounds for `-torsion in class groups obtained recently by Ellenberg and Venkatesh [EV], and subconvexit...
متن کاملOn the Central Derivative of Hecke L-series
The well-known Birch and Swinnerton-Dyer conjecture gives a deep connection between the leading coefficient of the L-series and the arithmetic properties of an abelian variety. Both are very important and subtle. This paper is part of an effort to compute the analytic side explicitly in a special case. Indeed, we are interested in the central derivative of certain algebraic Hecke L-series, rela...
متن کامل