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Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.

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Behaviour of (principal) polarizations of (singular) surfaces under birational maps

Assume we have two p.p. simple abelian surfaces $(A_i,D_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram: $\require{AMScd} \begin{CD} A_1 @>{birational}>> A_2\\ @V{2:1}VV @VV{2:1} …
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6 votes
1 answer
247 views

Are there curves of genus 2 with real multiplication by a non-maximal order?

Let us work over $\mathbb{C}$ for the moment. Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$. $\mathbf{Question:}$ Is there a smooth projective curve $C$ of gen …
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