Over the last two centuries mathematicians have developed a very rich theory of abelian varieties; however, certain kinds of explicit calculations have remained out of reach of our computational power until quite recently, when theoretical and technological advances have led to a renewed interest in the computational side of the theory. In this talk I will discuss one of the fundamental algorithmic problems one would like to solve, namely that of determining the endomorphism ring of the Jacobian of an explicitly given curve over a number field. I will describe a method to compute the endomorphism ring of such a Jacobian, starting with the case of genus-2 curves, and show the connection between this problem and a certain local-global principle for the center of the endomorphism algebra. If time permits, I will also outline how to prove the necessary local-global principle for abelian surfaces and, under the assumption of the Mumford-Tate conjecture, for all abelian varieties.
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