## Modular UnitsIn the present book, we have put together the basic theory of the units and cuspidal divisor class group in the modular function fields, developed over the past few years. Let i) be the upper half plane, and N a positive integer. Let r(N) be the subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex analytic isomorphic to an affine curve YeN), whose compactifi cation is called the modular curve X(N). The affine ring of regular functions on yeN) over C is the integral closure of C[j] in the function field of X(N) over C. Here j is the classical modular function. However, for arithmetic applications, one considers the curve as defined over the cyclotomic field Q(JlN) of N-th roots of unity, and one takes the integral closure either of Q[j] or Z[j], depending on how much arithmetic one wants to throw in. The units in these rings consist of those modular functions which have no zeros or poles in the upper half plane. The points of X(N) which lie at infinity, that is which do not correspond to points on the above affine set, are called the cusps, because of the way they look in a fundamental domain in the upper half plane. They generate a subgroup of the divisor class group, which turns out to be finite, and is called the cuspidal divisor class group. |

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### Contents

II | 1 |

III | 2 |

IV | 4 |

V | 8 |

VI | 11 |

VII | 12 |

VIII | 17 |

IX | 24 |

XLII | 159 |

XLIII | 165 |

XLIV | 168 |

XLV | 172 |

XLVI | 173 |

XLVII | 181 |

XLVIII | 186 |

XLIX | 190 |

X | 25 |

XI | 34 |

XII | 37 |

XIII | 42 |

XIV | 48 |

XV | 50 |

XVI | 58 |

XVII | 62 |

XVIII | 66 |

XIX | 68 |

XX | 75 |

XXI | 81 |

XXII | 84 |

XXIII | 87 |

XXIV | 90 |

XXV | 94 |

XXVI | 103 |

XXVII | 104 |

XXVIII | 110 |

XXIX | 111 |

XXX | 115 |

XXXI | 118 |

XXXII | 122 |

XXXIII | 126 |

XXXIV | 131 |

XXXV | 133 |

XXXVI | 140 |

XXXVII | 141 |

XXXVIII | 146 |

XXXIX | 147 |

XL | 151 |

XLI | 152 |

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abelian extension abelian group apply assume automorphism Cartan group Chapter character Cl(f class field coefficients complex multiplication concludes the proof condition congruence conjugates constant Corollary coset cuspidal divisor class cusps cyclotomic field define degree denominator denote distribution relations divides divisor class group elements elliptic curve equal expansion Fermat curve finite number follows formula Galois group group ring Hence homomorphism ideal class integral closure isomorphism Klein forms KUBERT lattice Let f modular curve modular forms modular function field modular units modulo N-th non-trivial notation number field polynomial positive integer power series prime number prime power prime to f primitive proof of Theorem proves the lemma quadratic relations right hand side roots of unity satisfies the quadratic Shimura Siegel functions Stickelberger ideal subgroup Suppose Tate curve Theorem 3.1 theory trivial unramified write

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Page 355 - Abh. 23 (1959) pp. 5-10 [Ya 1] K.. YAMAMOTO. The gap group of multiplicative relationships of Gaussian sums. Symposia Mathematica No. 15 (1975) pp. 427-440 [Ya 2] K. YAMAMOTO, On a conjecture of Hasse concerning multiplicative relations of Gaussian sums, J. Combin. Theory 1 (1966) pp.

Page 352 - K.. IWASAWA. A class number formula for cyclotomic fields. Ann. of Math.