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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
12
votes
Points of a variety defined by Galois descent
The following seems to give a reasonable affirmative answer which avoids computing the coordinate ring directly, and replaces condition (2) with the more natural condition that the subset $\Sigma := X …
15
votes
Existence of fine moduli space for curves and elliptic curves
If you want to work over a base ring such as $\mathbf{Z}[1/n]$ rather than over $\mathbf{Q}$ or $\mathbf{C}$ then the relevant numerical condition is that the part of $N$ coprime to $n$ not be "too sm …