Abstract: Abelian varieties are generalizations of elliptic curves into higher dimensions, carrying rich algebro-geometric as well as arithmetic information. We shall define what they are and give a couple important examples, along with some properties. The study of p-torsion of abelian varieties over a field of characteristic p is of particular interest, much less well understood than their elliptic curve counterparts. We will use this to motivate the study of the moduli spaces of certain classes of abelian varieties known as Shimura varieties.
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