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Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. ) Completely rewritten. …

Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller …

Yes! (I assume it was implicit in your question that the variety be projective?) More generally: any smooth, complex, rationally connected projective variety is simply connected. See Debarre's book …

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

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 …

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 …

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, …

On an abelian variety (regardless of the characteristic), an effective divisor with positive self-intersection is ample. To be more precise, it suffices here to recall that on any simple abelian varie …

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive work …

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 …

I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I wron …

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- …

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 …

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 …

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain a …