Department of

March 2015 April 2015 May 2015 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 1 2 8 9 10 11 12 13 14 5 6 7 8 9 10 11 3 4 5 6 7 8 9 15 16 17 18 19 20 21 12 13 14 15 16 17 18 10 11 12 13 14 15 16 22 23 24 25 26 27 28 19 20 21 22 23 24 25 17 18 19 20 21 22 23 29 30 31 26 27 28 29 30 24 25 26 27 28 29 30 31

Thursday, April 30, 2015

**Abstract:** I will present a new approach to several statistical questions about elliptic curves over finite fields, such as the average Lang-Trotter conjecture and the vertical Sato-Tate conjecture. The starting point is a theorem of Gekeler that provides a probabilistic reinterpretation of Deuring's theorem about the number of elliptic curves in a given isogeny class. In the heart of our approach lies a general technical theorem about averages of Euler products. As a corollary of this general result, we obtain new proofs of various results, some already known and some of which are new. One of the new results is the vertical Sato-Tate conjecture for very short intervals. This is joint work with Chantal David and Ethan Smith.