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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
5
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Does the moduli space of smooth curves of genus g contain an elliptic curve
If $g \geq 2$, and you want a map from the elliptic curve to the stack $M_g$ (as opposed to the coarse moduli space), then I think the map has to be constant, just on curvature grounds.