Elliptic Curves: Function Theory, Geometry, Arithmetic
Cambridge University Press, Aug 13, 1999 - Mathematics - 280 pages
The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, whose masters were Abel, Gauss, Jacobi, and Legendre. This book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows a historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. The many exercises with hints scattered throughout the text give the reader a glimpse of further developments. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics.
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1n short 1t follows absolute invariant algebraic arithmetic automorphism Check class invariants class number coefficients complex manifold complex structure computation conjugate constant coprime correspondence cubic cusp degree differential disk distinct divisor elliptic function elliptic integral equation of level example Exercise expressed fact field polynomial Figure finite fixed function field function of rational fundamental cell Galois group Gauss genus geometric ground field half-periods handlebody Hint identified identity invariant functions inverse irreducible j(co Jacobi Landen's transformation lattice modular equation modular function modular group modular substitution multiplicity null values p-function parameter period ratio permutations plane Platonic solids pole-free poles prime ideals produces projective line proved punctured quotient ramifications rational character rational function rational points Riemann surface root of unity simple roots single-valued sphere splitting field square-free Step subgroup theta functions torus universal cover upper half-plane upshot vanishes whole number