Elliptic curves of unbounded rank and Chebyshev's bias
type: Item
instance of: scholarly article
zbMATH Open document ID: 1323.11034
author: Daniel Fiorilli
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.8011
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 6365882
zbMATH Keywords: elliptic curves
zbMATH Keywords: rank over the rational field
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
MaRDI profile type: Publication
OpenAlex ID: W2963493224
arXiv ID: 1304.8011
DOI: 10.1093/IMRN/RNT103
published in: IMRN. International Mathematics Research Notices
title: Elliptic curves of unbounded rank and Chebyshev's bias
review text: It has been conjectured that there exist elliptic curves defined over \({\mathbb Q}\) with arbitrarily large rank. More precisely, there are two quantitative conjectures that relate the growth of the analytic rank of the elliptic curve to its conductor (by the way, incompatible with each other), made by \textit{D. Ulmer} [Math. Sci. Res. Inst. Publ. 49, 285--315 (2004; Zbl 1062.11033)] and by Farmer, Gonek and Hughes [\textit{D. W. Farmer} et al., J. Reine Angew. Math. 609, 215--236 (2007; Zbl 1234.11109)]. The author modifies these conjectures into a a weaker one (implied by both of them), namely the following ``analytic rank conjecture'': NEWLINE\[NEWLINE \limsup_{N_E \to +\infty} \frac {\text{r}_{\text{an}} (E)}{\sqrt {\log N_E}} = \infty, NEWLINE\]NEWLINE where \(N_E\) denotes the conductor of the elliptic curve \(E\) and \(\text{r}_{\text{an}} (E)\) denotes its analytic rank (i.e., the order of vanishing of \(L(E,s)\) at \(s=1\)). Defining, as usual, for a prime number \(p\), NEWLINE\[NEWLINE a_p(E)=p+1-\#E({\mathbb F})_p \;, NEWLINE\]NEWLINE it has also been observed that there is a bias in the distribution of \(a_p\), very much like in the distribution of primes congruent to 1 or to 3 modulo 4. The author considers the following sum, called the elliptic curve prime number race: NEWLINE\[NEWLINE S(t)= - \sum_ {p\leq t} \frac {a_p(E)}{\sqrt p}\, . NEWLINE\]NEWLINE Under certain hypotheses, including the Riemann hypothesis for elliptic curves, \textit{P. Sarnak} has shown that the function \(S(t)\) has in fact a bias [``Letter to Barry Mazur on `Chebyshev's bias' for \(\tau(p)\)'', \url{http://web.math.princeton.edu/sarnak/MazurLtrMay08.PDF}], and that it depends on the rank of the elliptic curve: more precisely, the limit NEWLINE\[NEWLINE \delta(E)=\lim_{T\to\infty} \frac 1{\log T} \int_{{2\leq t \leq T}\atop {S(t)\geq 0}} \frac {\text{d} t}tNEWLINE\]NEWLINE exists and it is \(<\frac 12 \) if \(\text{r}_{\text{an}} (E)=0\) and \(\geq \frac 12\) if \(\text{r}_{\…
reviewed by: Roberto Dvornicich
description: scientific article; zbMATH DE number 6365882