Modular Units

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Springer Science & Business Media, Sep 8, 1981 - Mathematics - 358 pages
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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
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XVII
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XVIII
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XIX
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XX
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XXI
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XXII
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XXIII
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XXIV
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XXV
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XXVI
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XXVII
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XXVIII
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XXIX
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XXX
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XXXI
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XXXII
122
XXXIII
126
XXXIV
131
XXXV
133
XXXVI
140
XXXVII
141
XXXVIII
146
XXXIX
147
XL
151
XLI
152
L
193
LI
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LII
211
LIII
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LIV
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LV
216
LVI
218
LVII
224
LVIII
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LIX
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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
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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.

References to this book

Cyclotomic Fields I and II
Serge Lang
No preview available - 1990
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About the author (1981)

Lang, Yale University, New Haven, CT.

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