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Let $A$ be an abelian variety over $\overline{\mathbb{Q}}$. We work up to isogeny (i.e., Hom sets are tensored with $\mathbb{Q}$). I am looking for a reference (and ideally, a short explanation) for t …

I am interested in the (degree $1$) Betti cohomology of the dual $A^\vee$ of an Abelian variety $A$ (say, over $\overline{\mathbb{Q}}$). We can even assume $A$ to be an elliptic curve, if this makes t …

I would like to ask for confirmation whether the following argument is correct. We work over an algebraically closed field $k$ of characteristic $0$. For an elliptic curve $E$, the Picard variety, or …

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over an algebraically closed field $k$ of characteristic zero. Throughout, let $x,y$ and $z$ be elements of $\mathfrak{g}$. The Baker-C …

I apologize in advance, since I am probably doing a very naive mistake in my computation. I am learning about pure (Chow / Grothendieck) motives. One of the first steps is to consider the category whe …

Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then $$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{ …

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence $$ 1 \to T \to G \to A \to 1$$ of algebraic groups, where $T$ is an algebraic …

I am reading the article by Lawrence and Venkatesh on diophantine problems and $p-$adic period mappings. At page $35$ they say that the dimension of the Prym variety of an (unramified) cover of curves …