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For an algebraic number $alpha$ and $gammain mathbb{R}$, $h(alpha)$ be the (logarithmic) Weil height, and $h_gamma(alpha)=(mathrm{deg}alpha)^gamma h(alpha)$ be the $gamma$-weighted (logarithmic) Weil height of $alpha$. Let $f:overline{mathbb{Q}}to [0,infty)$ be a function on the algebraic numbers $overline{mathbb{Q}}$, and let $Ssubset overline{mathbb{Q}}$. The Northcott number $mathcal{N}_f(S)$ of $S$, with respect to $f$, is the infimum of all $Xgeq 0$ such that ${alpha in S; f(alpha)< X}$ is infinite. This paper studies the set of Northcott numbers $mathcal{N}_f(mathcal{O})$ for subrings of $overline{mathbb{Q}}$ for the house, the Weil height, and the $gamma$-weighted Weil height. We show: (1) Every $tgeq 1$ is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each $tgeq 0$ there exists a field with Northcott number in $ [t,2t]$ w.r.t. the Weil height $h(cdot)$. (3) For all $0leq gammaleq 1$ and $gamma<gamma$ there exists a field $K$ with $mathcal{N}_{h_{gamma}}(K)=0$ and $mathcal{N}_{h_gamma}(K)=infty$. For $(1)$ we provide examples that satisfy an analogue of Julia Robinons property (JR), examples that satisfy an analogue of Vidaux and Videlas isolation property, and examples that satisfy neither of those. Item $(2)$ concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.
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We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce certain height and rank estimates on abelian varieties over a number field $K$ to the case of jacobian varieties defined over a suitable extension of $K$.
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We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=sum_{x in F} psi(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $psi$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo $|F^times|$ (degenerate $d$), which yield trivial Weil sums. We prove that $V_{F,d} leq (2/3)[Fcolon{mathbb F}_p]$ for any $F$ and any nondegenerate $d$, and prove that this bound is actually reached in infinitely many fields $F$. We also prove some stronger bounds that apply when $[Fcolon{mathbb F}_p]$ is a power of $2$ or when $d$ is not congruent to $1$ modulo $p-1$, and show that each of these bounds is reached for infinitely many $F$.
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