Abstract
Supplementing other treatments, this article discusses the history and meaning of four theorems that have been accepted as Abel’s Theorem. The discussion explains Abel’s own proofs, and pays due attention to the hyperelliptic case. Section 1 explains how each theorem came to be called Abel’s Theorem. Section 2 treats Abelian integrals, Clebsch’s geometric reformulation, and Abel’s Elementary Function Theorem. Section 3 treats Abel’s Equivalence Theorem, the genus, and adjoints. Section 4 treats the Riemann—Roch Theorem and Abel’s Relations Theorem. Section 5 explains the various forms of Abel’s Addition Theorem and Abel’s proofs of them. Section 6 discusses the Abel map, and uses it to prove the Addition Theorem in its most elaborate form. Section 7 discusses the Picard and Albanese varieties, and explains Abel’s version of the genus. Section 8 sums it all up, and concludes that only the Addition Theorem can rightfully be called Abel’s Theorem.
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