Elliptic curves of unbounded rank and Chebyshev's bias
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 \(\text{r}_{\text{an}} (E)>0\).NEWLINENEWLINELet \(\underline\delta\) and \(\overline\delta\) be the \(\liminf\) and the \(\limsup\) in the preceding formula. Based on these observations, the author shows, again under some hypotheses, that the analytic rank conjecture implies the existence of elliptic curves such that \(1-\varepsilon < \underline\delta(E)\leq \overline\delta(E)<1\) for each \(\varepsilon >0\), and so that the bias can be as large as possible. Other assumptions allow the author to prove that if there is a sequence \(E_n\) of elliptic curves for which \(\bar\delta (E_n)\) tends to 1, then the analytic rank conjecture holds.
scientific article; zbMATH DE number 6365882