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1. CJM Online first
The frequency of elliptic curve groups over prime finite fields Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is well-known that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4-\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.
Keywords:average order, elliptic curves, primes in short intervals Categories:11G07, 11N45, 11N13, 11N36 |
2. CJM 2012 (vol 64 pp. 1019)
On a Theorem of Bombieri, Friedlander, and Iwaniec In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.
Keywords:primes in arithmetic progressions, Titchmarsh divisor problem Category:11N13 |