Abstract
Elliptic curves are algebraic groups in addition to being algebraic curves; a modular form classifies elliptic curves by assigning a value in a ring A to an elliptic curve with some extra structures defined over A. From a different viewpoint, automorphic forms are generally defined on the adele points of a linear algebraic group; in particular, modular forms are functions on \(GL_{2}(\mathbb{A})\) (e.g., (AAG)). Thus, a minimal amount of knowledge of group schemes is necessary to describe elliptic curves and modular forms. Group schemes are best understood from the functorial viewpoint of scheme theory, regarding a scheme as a functor from the category of algebras to sets taking each algebra R to the set of R-rational points S(R) of the scheme S.
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