# Modular Units

Springer Science & Business Media, Sep 8, 1981 - Mathematics - 358 pages
In 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
 L 193 LI 197 LII 211 LIII 213 LIV 214 LV 216 LVI 218 LVII 224 LVIII 227 LIX 229 LX 233 LXI 241 LXII 246 LXIII 252 LXIV 260 LXV 266 LXVI 269 LXVII 277 LXVIII 285 LXIX 298 LXX 303 LXXI 311 LXXII 317 LXXIII 321 LXXIV 323 LXXV 327 LXXVI 329 LXXVII 339 LXXVIII 351 LXXIX 357 Copyright

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