THE MUSIC OF THE PRIMES

By AMELIA WILSON

CONTENTS INTRODUCTION TWO SUBJECTS THAT BRANCHED OUT FROM C F GAUSS—QUADRATIC RESIDUES AND LINKING NUMBERS GEOMETRIZATION OF NUMBER THEORY THE OUTLINE OF THIS BOOK PRELIMINARIES: FUNDAMENTAL GROUPS AND GALOIS GROUPS THE CASE OF TOPOLOGICAL SPACES THE CASE OF ARITHMETIC RINGS ARITHMETIC DUALITY THEOREMS 2 3 1 FINITE FIELDS 2 3 2 P-ADIC FIELDS 2 3 3 NUMBER RINGS KNOTS AND PRIMES, 3-MANIFOLDS AND NUMBER RINGS LINKING NUMBERS AND LEGENDRE SYMBOLS LINKING NUMBERS LEGENDRE SYMBOLS DECOMPOSITIONS OF KNOTS AND PRIMES DECOMPOSITION OF A KNOT DECOMPOSITION OF A PRIME HOMOLOGY GROUPS AND IDEAL CLASS GROUPS I: GENUS THEORY HOMOLOGY GROUPS AND IDEAL CLASS GROUPS GENUS THEORY FOR A LINK GENUS THEORY FOR PRIMES IDELIC CLASS FIELD THEORY FOR 3-MANIFOLDS AND NUMBER RINGS STABLY GENERIC LINKS AND CHEBOTAREV LINKS IDELIC CLASS FIELD THEORY FOR 3-MANIFOLDS IDELIC CLASS FIELD THEORY FOR NUMBER FIELDS LINK GROUPS AND GALOIS GROUPS WITH RESTRICTED RAMI?CATION LINK GROUPS PRO-L GALOIS GROUPS WITH RESTRICTED RAMI?CATION MILNOR INVARIANTS AND MULTIPLE POWER RESIDUE SYMBOLS FOX FREE DIFFERENTIAL CALCULUS MILNOR INVARIANTS PRO-L FOX FREE DIFFERENTIAL CALCULUS MULTIPLE POWER RESIDUE SYMBOLS ALEXANDER MODULES AND IWASAWA MODULES 10 1 DIFFERENTIAL MODULES 10 2 THE CROWELL EXACT SEQUENCE 10 3 COMPLETE DIFFERENTIAL MODULES 10 4 THE COMPLETE CROWELL EXACT SEQUENCE HOMOLOGY GROUPS AND IDEAL CLASS GROUPS II: HIGHER-ORDER GENUS THEORY 11 1 THE UNIVERSAL LINKING MATRIX FOR A LINK 11 2 HIGHER-ORDER GENUS THEORY FOR A LINK 11 3 THE UNIVERSAL LINKING MATRIX FOR PRIMES 11 4 HIGHER ORDER GENUS THEORY FOR PRIMES HOMOLOGY GROUPS AND IDEAL CLASS GROUPS III: ASYMPTOTIC FORMULAS 12 1 THE ALEXANDER POLYNOMIAL AND HOMOLOGY GROUPS 12 2 THE IWASAWA POLYNOMIAL AND P-IDEAL CLASS GROUPS TORSIONS AND THE IWASAWA MAIN CONJECTURE 13 1 TORSIONS AND ZETA FUNCTIONS 13 2 THE IWASAWA MAIN CONJECTURE MODULI SPACES OF REPRESENTATIONS OF KNOT AND PRIME GROUPS 14 1 CHARACTER VARIETIES OF COMPLEX REPRESENTATIONS OF A KNOT GROUP 14 2 THE CHARACTER VARIETY OF COMPLEX 1-DIMENSIONAL REPRESENTATIONS OF A KNOT GROUP AND ALEXANDER IDEALS 14 3 UNIVERSAL DEFORMATION SPACES OF P-ADIC REPRESENTATIONS OF A PRIME GROUP 14 4 THE UNIVERSAL DEFORMATION SPACE OF P-ADIC 1-DIMENSIONAL REPRESENTATIONS OF A PRIME GROUP AND IWASAWA IDEALS DEFORMATIONS OF HYPERBOLIC STRUCTURES AND P-ADIC ORDINARY MODULAR FORMS 15 1 DEFORMATION OF HYPERBOLIC STRUCTURES 15 2 DEFORMATION OF P-ADIC ORDINARY MODULAR GALOIS REPRESENTATIONS DIJKGRAAF–WITTEN THEORY FOR 3-MANIFOLDS AND NUMBER RINGS 16 1 CHERN–SIMONS FUNCTIONALS AND DIJKGRAAF–WITTEN INVARIANTS FOR CLOSED 3-MANIFOLDS 16 2 DIJKGRAAF–WITTEN TQFT FOR 3-MANIFOLDS WITH BOUNDARIES 16 3 ARITHMETIC CHERN–SIMONS FUNCTIONALS AND ARITHMETIC DIJKGRAAF–WITTEN INVARIANTS FOR NUMBER FIELDS 16 4 ARITHMETIC DIJKGRAAF–WITTEN TQFT FOR S-INTEGER NUMBER RINGS