Heights and totally real numbers

1973 Schinzel proved that the standard logarithmic height h on the maximal totally real field extension of the rationals is either zero or bounded from below by a positive constant. In this paper we study this property for canonical heights associated to rational functions and the corresponding dynamical system on the affine line. At the end, we will give a few remarks on the behavior of h on finite extensions of the maximal totally real field.Comment: Major changes regarding the first version. E.g. the last chapter was cancele

Heights and totally pp-adic numbers

We study the behavior of canonical height functions $\widehat{h}_f$, associated to rational maps $f$, on totally $p$-adic fields. In particular, we prove that there is a gap between zero and the next smallest value of $\widehat{h}_f$ on the maximal totally $p$-adic field if the map $f$ has at least one periodic point not contained in this field. As an application we prove that there is no infinite subset $X$ in the compositum of all number fields of degree at most $d$ such that $f(X)=X$ for some non-linear polynomial $f$. This answers a question of W. Narkiewicz from 1963.Comment: minor changes: rewording and reference update

Fields with the Bogomolov property

A field is said to have the Bogomolov property relative to a height function h’ if and only if h’ is either zero or bounded from below by a positive constant for all for all elements in this field. In this thesis we study this property according to canonical heights associated to rational functions introduced by Call and Silverman in 1994. In the first part we will translate known results into the dynamical setting. Then we prove an effective version of a theorem of Baker which states that the Néron-Tate height of an elliptic curve with multiplicative reduction at a finite place v is bounded from below by a positive constant at points which are unramified over v. In the last section of this thesis we give a complete classification of rational functions f defined over the algebraic numbers such that the maximal totally real field has the Bogomolov property relative to the canonical height associated to f

News on quadratic polynomials

Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning quadratic polynomials, which uses a bridge between algebra and analysis. We study the iterations of quadratic polynomials, obtained by computing the value of a polynomial for a given number and feeding the outcome into the exact same polynomial again. These iterations of polynomials have interesting applications, such as in fractal theory

Heights of points with bounded ramification

Let $E$ be an elliptic curve defined over a number field $K$ with fixed non-archimedean absolute value $v$ of split-multiplicative reduction, and let $f$ be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on $E$ is either zero or bounded from below by a positive constant, for all points of bounded ramification over $v$. In this paper we make this bound effective and prove an analogue result for the canonical height associated to $f$. We also study variations of this result by changing the reduction type of $E$ at $v$. This will lead to examples of fields $F$ such that the N\'eron-Tate height on non-torsion points in $E(F)$ is bounded from below by a positive constant and the height associated to $f$ gets arbitrarily small on $F$. The same example shows, that the existence of such a lower bound for the N\'eron-Tate height is in general not preserved under finite field extensions.Comment: Major changes regarding the first version. E.g.: the title was changed; errors were corrected; based on a remark of Joseph Silverman Example 5.7 and Theorem 5.9 were adde