Elliptic Curves: Function Theory, Geometry, ArithmeticThe 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. |
Contents
II | 1 |
III | 3 |
IV | 7 |
V | 8 |
VI | 12 |
VII | 14 |
VIII | 16 |
IX | 24 |
XLII | 154 |
XLIII | 159 |
XLIV | 160 |
XLV | 162 |
XLVI | 166 |
XLVII | 167 |
XLVIII | 169 |
XLIX | 172 |
X | 27 |
XI | 29 |
XII | 33 |
XIII | 46 |
XIV | 51 |
XV | 52 |
XVI | 54 |
XVII | 55 |
XVIII | 62 |
XIX | 65 |
XX | 68 |
XXI | 71 |
XXII | 77 |
XXIII | 81 |
XXIV | 84 |
XXV | 87 |
XXVI | 89 |
XXVII | 92 |
XXVIII | 93 |
XXIX | 98 |
XXX | 104 |
XXXI | 109 |
XXXII | 125 |
XXXIII | 127 |
XXXIV | 131 |
XXXV | 133 |
XXXVI | 135 |
XXXVII | 140 |
XXXVIII | 142 |
XXXIX | 143 |
XL | 147 |
XLI | 151 |
Other editions - View all
Elliptic Curves: Function Theory, Geometry, Arithmetic Henry McKean,Victor Moll Limited preview - 1999 |
Common terms and phrases
absolute invariant algebraic arithmetic automorphism Check class invariants class number coefficients complex manifold complex structure computation conformal conjugate constant coprime cubic cusp degree differential disk divisor e₁ elliptic function elliptic integral equation of level example Exercise fact field polynomial Figure finite fixed function field fundamental cell Galois group Gauss genus geometric ground field half-periods handlebody Hint identified identity imaginary inverse irreducible j(pk Jacobi's K₁ k²x² Landen's transformation lattice modular equation modular group modular substitution multiplicity null values parameter period ratio permutations plane Platonic solids poles produces projective line proved punctured quadratic quotient ramifications rational character rational function rational points Riemann surface Section simple roots single-valued sphere splitting field Step subgroup torus universal cover upper half-plane vanishes w₁ whole number x₁ г₁