Chantal David, Concordia University
“Averages of Euler products and statistics of elliptic curves
Joint work with D. Koukoulopoulos and E. Smith.”
We present in this talk several results related to statistics of elliptic curves over a finite field Fp, which follow from a general theorem about averages of Euler products. In this general framework, we can reprove known results as the vertical Lang-Trotter conjecture, the vertical Koblitz conjecture, and the vertical Sato-Tate conjecture (for very short intervals). We can also compute statistics for new questions as the problem of amicable pairs and aliquot cycles,
First introduced by Silverman and Stange. Our technique is broad and general, and easily appli- cable to other distribution questions. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring’s theorem in terms of random matrix theory, making a direct con- nection between the (conjectural) horizontal distributions and the vertical distributions. A key step of our results is to control the stabilization of the Euler factors appearing in Gekeler’s theorem: for each ‘, the Euler factor related to random matrix theory is defined as the r-limit of a matrix count modulo ‘r, which stabilizes for some r depending on ‘ and p. Our main result shows that under cer- tain conditions for this stabilization, a weighted average of Euler products is asymptotic to the Euler product of the average factors (which are stable for r = 1).