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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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Is it possible to classify all Weil cohomologies?

The answer to this question is yes. Roughly speaking: make a category Weil(A) of Weil cohomologies with values in a rigid tensor abelian category A and let Weil(-) be the resulting 2-functor. Then Wei …
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