Explicit formulae and the Lang-Trotter conjecture
type: Item
instance of: scholarly article
title: Explicit formulae and the Lang-Trotter conjecture
zbMATH Open document ID: 0587.14009
DOI: 10.1216/RMJ-1985-15-2-535
author: V. Kumar Murty
published in: Rocky Mountain Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over \(\mathbb Q\) of conductor \(N\). Define \[ \pi_ E(x)=\#\{p<x: p\nmid N\text{ and } N_ p=p+1\} \] where \(N_ p\) is the number of points of the reduction of \(E \mod p\) defined over the field of \(p\) elements. Let \(L_ k(s)\) be the ``\(k\)-th symmetric power'' of the \(L\)-series attached to \(E\). Assuming analytic continuation, functional equation and Riemann hypothesis, the author studies by analytic methods the integral \[ \frac 1{2\pi i}\int^{c+iT}_{c- iT}(L'/L)(s) x^ s \,\frac{ds}{s} \] and deduces for example that if \(E\) does not have complex multiplication, then \[ \pi_ E(x)\ll x^{3/4}(\log Nx)^{1/2}. \]
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 3941675
zbMATH Keywords: symmetric power of L-series
zbMATH Keywords: elliptic curve
zbMATH Keywords: analytic continuation
zbMATH Keywords: functional equation
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: complex multiplication
Wikidata QID: Q123358522
MaRDI profile type: Publication
full work available at URL: rmj-1985-15-2-535
OpenAlex ID: W2159596114
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description: scientific article; zbMATH DE number 3941675