2019 Oregon Number Theory Days Spring Meeting

The spring meeting of Oregon Number Theory Days will take place on Saturday, June 1 at Portland State University. This is a triannual number theory seminar rotating between Oregon State University, Portland State University, and the University of Oregon.

Speakers:

Chantal David (Concordia University)

Neha Prabhu (Queens University)

Program:

10:00-11:00 -- Chantal David (Concordia University), Lecture I: Explicit Sato-Tate conjecture and other equidistribution conjectures for elliptic curves

11:20-11:50 -- Neha Prabhu (Queen's University): The error term in the Sato-Tate law of Birch

12:00-2:00 -- Lunch

2:00-2:50 -- Chantal David (Concordia University), Lecture II: Distributions for elliptic curves over finite fields

Abstracts:

• Chantal David, Lecture I: Explicit Sato-Tate conjecture and other equidistribution conjectures for elliptic curves

We present in this talk several natural prime counting conjectures associated to elliptic curves over Q, concerning the reductions of a fixed elliptic curve E over the finite fields Fp, varying over all primes p. Among such conjectures is the Sato-Tate conjecture, which was proven recently by Taylor, in collaboration with Clozel, Harris and Shepherd-Barron. His breakthrough work proves a non-effective version of the Sato-Tate conjecture. We explain in this talk how assuming more hypotheses leads to an effective version of the Sato-Tate conjecture, and how this effective version of the Sato-Tate conjecture gives upper bounds for other prime counting conjectures associated to elliptic curves over Q which are still open, as the Lang-Trotter conjecture, or the extremal primes conjecture. This talk is based on joint work with A. Gafni, A. Malik, N. Prabhu and C. Turnage-Butternaugh.

• Chantal David, Lecture II: Distributions for elliptic curves over finite fields

We present several distribution results for elliptic curves over finite fields Fp which follow from a general result about averages of weighted Euler products. In this general framework, we can reprove known results such as vertical analogues of the Lang-Trotter conjecture, the Koblitz conjecture, and the Sato-Tate conjecture, even for very short intervals which were not accessible by previous methods. We can also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. We will explain how the starting point of our results is a theorem of Gekeler which expresses the previously known formulas for the number of elliptic curves over Fp with a given number of point in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. This talk is based on joint work with D. Koukoulopoulos and E. Smith.

• Neha Prabhu: The error term in the Sato-Tate law of Birch

In 1968, Birch proved a vertical analogue of the Sato-Tate conjecture for elliptic curves. For p prime, he investigated the asymptotic behaviour of aE(p), the second order term in the expression for the number of points on E defined mod p, as p grows. An error term for this result was obtained by Banks and Shparlinski in 2009 using the work of Katz and Deligne. In this talk, we shall see that this error term can also be obtained in an elementary fashion using ideas in Birch's paper. This is joint work with Ram Murty.