Mathematics > Number Theory
[Submitted on 20 Oct 2015 (v1), last revised 26 Sep 2016 (this version, v3)]
Title:Sums of Euler products and statistics of elliptic curves
View PDFAbstract:We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang-Trotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.
Submission history
From: Dimitris Koukoulopoulos [view email][v1] Tue, 20 Oct 2015 15:45:00 UTC (46 KB)
[v2] Sun, 31 Jan 2016 20:55:30 UTC (47 KB)
[v3] Mon, 26 Sep 2016 16:43:55 UTC (48 KB)
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