2013 CMS Summer Meeting
Dalhousie University, June 4 - 7, 2013
Krieger-Nelson Prize Lecture
- CHANTAL DAVID, Concordia University
Average distributions for elliptic curves [PDF]
-
Let $E$ be an elliptic curve over ${\bf Q}$ .
There are many open conjectures about the distribution of local invariants associated with
the reductions of $E$ modulo $p$ as $p$ varies over the primes.
Perhaps the most famous examples are the conjectures of Lang and Trotter (1976) and
Koblitz (1988).
In order to gain evidence for the conjectures, it is natural to
consider the average distribution over some families of elliptic curves. We explain
how the average results fit the conjectural asymptotics, in terms of the order of magnitude, but
also in terms of the precise constants associated to each given conjecture, giving evidence
for the probabilistic models defined in terms
of local probabilities. More recently, applying those average techniques to different distribution questions, as counting the frequency of occurrence of a given abelian
group appearing as the group of points of elliptic curves over finite fields,
we found that the resulting average distribution is also
governed by the Cohen-Lenstra Heuristics, which
predict that random abelian groups occur with probability
weighted by the number of elements of their automorphism group.
