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Abelian Varieties - James Milne

Abelian MilneVersion 16, 2008 These notes are an introduction to the theory of Abelian Varieties , including the arithmeticof Abelian Varieties and Faltings s proof of certain finiteness theorems. The orginal versionof the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough information@misc{milneAV,author={ Milne , James S.},title={ Abelian Varieties ( )},year={2008},note={Available at },pages={166+vi}} (July 27, 1998). First version on the web, 110 (March 17, 2008).

Introduction The easiest way to understand abelian varieties is as higher-dimensional analogues of ellip-tic curves. Thus we first look at the various definitions of an elliptic curve. Fix a ground field kwhich, for simplicity, we take to be algebraically closed. An elliptic curve over k can be defined, according to taste, as:

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