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We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.
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In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a different approach to the uniform Mordell-Lang type of result of Dimitrov-Gao-Habegger and Kuhne. The treatment is based on the recent theory of adelic line bundles of Yuan-Zhang.
Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.
Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $bar{x}$ with $0<bar{x}<q$ such that $xbar{x}equiv1(bmod q)$. A Lehmer number is defined to be any integer $a$ with $2dagger(a+bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=displaystylemathop {displaystylemathop{sum{}}_{a=1}^{q} displaystylemathop{sum{}}_{bleq xq}}_{mbox{$tinybegin{array}{c} 2|a+b+1 abequiv1(bmod q)end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymptotic formula, by using estimates of Kloostermans sums and properties of trigonometric sums.
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.
In this paper, we extend the notion of the Bogomolov multipliers and the CP-extensions to Lie algebras. Then we compute the Bogomolov multipliers for Abelian, Heisenberg and nilpotent Lie algebras of class at most 6. Finally we compute the Bogomolov multipliers of some simple complex Lie algebras.
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