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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 …

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 …

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, …

Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $ …

In Colmez's formulation, it is necessary that the endomorphism ring be the maximal order $\mathcal{O}_k$. It is then proved only in special cases ($k/\mathbb{Q}$ abelian), or on average over the CM ty …

Stepanov (circa 1970) created the polynomial method to limit the rational points of an algebraic curve over $\mathbb{F}_q$, leading to one of several alternative proofs of Weil's Riemann hypothesis fo …

In Arakelov geometry, the conventional wisdom is that the ``closed fibre at $\infty$'' should be viewed as totally degenerate. This is the extreme opposite of smoothness. A visualization in the case o …

All accumulation points of $J_p$ in $\mathbb{C}_p$ are roots of degree two monic equations over $\mathbb{Z}_p$, and their approximants are necessarily supersingular at $p$. Moreover, there exist accum …

All such resultants and/or discriminants are geometrically irreducible in characteristic zero, and a power of an irreducible in general. This is actually covered by the geometric argument I quoted in …

The discriminant locus has the following geometric interpretation, given in the introductory chapter of [Gelfand, Kapranov, Zelevinsky: Discriminants, Resultants and Multidimensional Determinants]. L …

Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser a …

You mean the six points to be distinct, of course (or not all six points to be the same point). Fixing the analytic identification $(\wp(z),\wp'(z))$ with $T = \mathbb{C}/\mathbb{\Lambda}$, the Abel- …

This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2 …

Binary quadratic forms arise in nature as norm forms for a quadratic field. This point of view has various consequences in number theory. For a fixed negative discriminant (the definite case), Gauss …

Will Sawin and Michael Stoll have noted that, as a consequence of Faltings's "Big Theorem," a hyperelliptic equation $y^2 = f(x)$ with $\deg{f} > 6$ (genus $> 2$) and not admitting a degree $2$ non-co …