LIMIT THEOREMS FOR TWISTS OF L-FUNCTIONS OF ELLIPTIC CURVES. II

Virginija Garbaliauskienė,Antanas Laurinčikas

doi:10.3846/13926292.2012.645170

Virginija Garbaliauskienė, Antanas Laurinčikas

Open Access

https://doi.org/10.3846/13926292.2012.645170

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Abstract

In the paper, a limit theorem for the argument of twisted with Dirichlet character L-functions of elliptic curves with an increasing modulus of the character is proved.

Highlights

IntroductionL-functions of elliptic curves with an increasing modulus of the character, and obtained a limit theorem of such a type for the modulus of these twists
In [3], we began to study limit theorems for twisted with Dirichlet characterL-functions of elliptic curves with an increasing modulus of the character, and obtained a limit theorem of such a type for the modulus of these twists
Denote by Ep the reduction of the curve E modulo p which is a curve over the finite field Fp, and define λ(p) by

Summary

Introduction

L-functions of elliptic curves with an increasing modulus of the character, and obtained a limit theorem of such a type for the modulus of these twists. |E(Fp)| = p + 1 − λ(p), where |E(Fp)| is the number of points of Ep. The L-function LE(s), s = σ + it, of the elliptic curve E is defined by the Euler product λ(p) −1 λ(p). The twist LE(s, χ) with Dirichlet character χ for the function LE(s) is defined . Let aτ (m) and bτ (m), m ∈ N, be multiplicative functions defined by (1.4)–(1.6), i.e., aτ (m) =. Converges weakly mod 1 to the distribution function mod 1 defined by the Fourier transform g(k) as Q → ∞. We can prove Theorem 2 only in the half-plane of absolute convergence of the mentioned series. We have a conjecture that the statement of Theorem 2 remains true for σ > 1, at the moment we can not prove this

Fourier Transform

Proof of Theorem 2

Findings

MQ q Q