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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 Online first
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials Assuming Lang's conjectured lower bound on the heights of non-torsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion
point $P \in E(\mathbb{Q})$, there is at most one integral multiple
$[n]P$ such that $n \gt C$. The proof is a modification of a proof
of Ingram giving an unconditional but not uniform bound. The
new ingredient is a collection of explicit formulae for the
sequence $v(\Psi_n)$ of valuations of the division polynomials.
For $P$ of non-singular reduction, such sequences are already
well described in most cases, but for $P$ of singular reduction,
we are led to define a new class of sequences called \emph{elliptic
troublemaker sequences}, which measure the failure of the NÃ©ron
local height to be quadratic. As a corollary in the spirit of
a conjecture of Lang and Hall, we obtain a uniform upper bound
on $\widehat{h}(P)/h(E)$ for integer points having two large
integral multiples.
Keywords:elliptic divisibility sequence, Lang's conjecture, height functions Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52 |
3. CJM 2004 (vol 56 pp. 673)
DÃ©faut de semi-stabilitÃ© des courbes elliptiques dans le cas non ramifiÃ© Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified
finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic
curve defined over $K$ with additive reduction over $K$, and having an integral
modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of
$K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$
of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the
degree of the extension $L/K_{nr}$.
Category:11G07 |