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I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly background;...

Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have: a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$ a finitely-...