Elliptic Functions and Elliptic Integrals
American Mathematical Soc., Sep 16, 1997 - Mathematics - 185 pages
This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.
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Abel's theorem addition of points addition theorem arc length assume automorphism calculations change of parameter change of variables coefficients complex numbers construct coordinates corresponds cubic curve cubic equation curve Sp curve y2 degree equation degree polynomial distinct divisible divisor easy to verify elliptic curve elliptic function elliptic integrals equal equation y2 expressed in terms Figure finite follows form y2 formula function j(r fundamental parallelogram Hence infinite point inflection points integer solution intersection points irreducible irreducible polynomial isomorphism k2 sin2 lattice lemniscate Let us consider Let us prove Let us show linear mathematician meromorphic Moreover non2ero nonsingular cubic obtained parameteri2ation problem projective plane proof quintic equation radicals rational function rational numbers rational point rational solutions relation relatively prime root of unity ruler and compass Serret's curves singular points solvable straight line subgroup suffices tangent theory transformation values vanish Weierstrass function