Algebraic Definition
Two equivalent definitions of abelian variety over a general field k are commonly in use:
- a connected and complete algebraic group over k
- a connected and projective algebraic group over k.
When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curves are abelian varieties of dimension 1.
In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second. Only in 1948 did he prove that complete algebraic groups can be embedded into projective space. Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the Algebraic Geometry article).
Read more about this topic: Abelian Variety
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