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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

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 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 functionsCategories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52

2. CJM 2004 (vol 56 pp. 673)

Cali, Élie
 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