Algebraic Geometry
There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus. When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points). The definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.
Read more about this topic: Genus (mathematics)
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