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abx's comment was made while I was writing this, but I am posting it as an answer anyway. There has not been a proof of this conjecture of Lang, which remains a wide open problem. Lu and Miyaoka's pa …

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and …

There should be better references, but for a beginning, how about the following: Look at Lemme 1 in Thierry Bouche's paper Convergence de la metrique de Fubini-Study d'un fibre lineaire positif, Ann. …

I do not know if this falls within the scope of your question, and moreover I do not have a specific reference to point to, but there are certainly plenty of unsolved questions involving the dynamics …

Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields unram …

As the night sky, mathematics has two hemispheres; the archimedean hemisphere and the non-archimedean hemisphere. For some reasons, the latter hemisphere is usually under the horizon of our world, …

As you indicate, real algebraic numbers of degree $\leq 2$ have this property in view of Lagrange's classical result characterizing them by the eventual periodicicty of the continued fractions expansi …

This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIM …

This is true; it is due to Selmer. Ljunggren (On the irreducibility of certain trinomials and quadrinomials, Math. Scand. 1960) has obtained the complete list of reducible trinomials with $\pm 1$ coef …

Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer. Amoroso and Dvornicich discovered (A lower bound on the height in abelian ext …