Limit Theorems for Twists of L-Functions of Elliptic Curves. Iv

Abstract

In this paper, we prove a multidimensional limit theorem for moduli of twists of L-functions of elliptic curves. The limit measure in this theorem is defined by the characteristic transforms.

A Joint Elliott Type Theorem for Twists of L-Functions of Elliptic Curves

“…uniformly in σ ∈ K and q. Moreover, in view of the equalities 8) and the bound d(m) m ε , ε > 0, we obtain that the second term in the right-hand side of (3.4) has a bound…”

“…Then it was observed in [12] that the function L E (s, χ), as L E (s), is also entire one. Limit theorems for L E (s, χ) with increasing q were began to study in [6], [7], [8], [9], however, only in the half-plane of absolute convergence σ > 3 2 . In [12], a limit theorem in the space of analytic functions H(D), D = {s ∈ C : σ > 1}, for the function L E (s, χ) has been obtained.…”

“…In [8], a joint limit theorem of type of Theorem 1 has been obtained for a collection of moduli of twists of L-functions of elliptic curves, however, in the region σ > 3 2 , only. We note that P. D. T. A. Elliott was the first who began to study limit theorems with increasing modulus for Dirichlet L-functions [4], [5].…”

Теорема Типа Эллиотта Для Сверток $L$-Функций Эллиптических Кривых

“…когда → ∞, слабо сходится к предельной мере̂︀ на (C, ℬ(C)), определяемой характеристическим преобразованием̂︀( , ). В статьях [6]- [8] были получены предельные теоремы типа Эллиотта для сверток -функций эллиптических кривых с характером Дирихле, однако только в области абсолютной сходимости ряда, которым определяется -функция. Пусть эллиптическая кривая над полем рациональных чисел, заданная уравнением Вейерштрасса…”