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Search: MSC category 11G07 ( Elliptic curves over local fields [See also 14G20, 14H52] )

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1. CJM Online first

Chandee, Vorrapan; David, Chantal; Koukoulopoulos, Dimitris; Smith, Ethan
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

Stange, Katherine E.
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 Néron 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)

Cali, Élie
Défaut de semi-stabilité des courbes elliptiques dans le cas non ramifié
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 $L/K_{nr}$.


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