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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
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Tower of moduli spaces in Scholze's theory
This is more of a comment, but since I'm not allowed to comment I'll do it here.
Firstly, this moduli interpretation is mentioned in Scholze, look at (for example) Lemma III.3.4 in his torsion paper …