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1. CJM 2015 (vol 67 pp. 1046)

Dubickas, Arturas; Sha, Min; Shparlinski, Igor
 Explicit Form of Cassels' $p$-adic Embedding Theorem for Number Fields In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$. Keywords:number field, $p$-adic embedding, height, polynomial, cyclotomic fieldCategories:11R04, 11S85, 11G50, 11R09, 11R18

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 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

3. CJM 2011 (vol 64 pp. 301)

Hurlburt, Chris; Thunder, Jeffrey Lin
 Hermite's Constant for Function Fields We formulate an analog of Hermite's constant for function fields over a finite field and state a conjectural value for this analog. We prove our conjecture in many cases, and prove slightly weaker results in all other cases. Category:11G50

4. CJM 2009 (vol 61 pp. 1118)

Pontreau, Corentin
 Petits points d'une surface Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \a ce probl\eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\etement explicites, peuvent toutefois s'\'etendre \a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \`a la dimension $n$. Keywords:Hauteur normalisÃ©e, groupe multiplicatif, problÃ¨me de Lehmer, petits pointsCategories:11G50, 11J81, 14G40

5. CJM 2008 (vol 60 pp. 1406)

Ricotta, Guillaume; Vidick, Thomas
 Hauteur asymptotique des points de Heegner Geometric intuition suggests that the N\'{e}ron--Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross--Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin--Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves. Categories:11G50, 11M41

6. CJM 2007 (vol 59 pp. 1284)

Fukshansky, Lenny
 On Effective Witt Decomposition and the Cartan--Dieudonn{Ã© Theorem Let $K$ be a number field, and let $F$ be a symmetric bilinear form in $2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical theorem of Witt states that the bilinear space $(Z,F)$ can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights of $F$ and $Z$. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'e} theorem. Namely, we show that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights of $F$, $Z$, and $\sigma$. Keywords:quadratic form, heightsCategories:11E12, 15A63, 11G50
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