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Weber's class invariants

Published online by Cambridge University Press:  26 February 2010

B. J. Birch
B. J. Birch
Affiliation:
Mathematical Institute, 24-29 St. Giles, Oxford.
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Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular invariant j(w); he calls j(w) a class invariant. He goes on in §§125-144 to consider the values f(w) of other modular functions f(z); he shows that in certain cases the degree of the extension ℚ(ω, f(ω)) of ℚ(ω, j(ω)) is much less than that of ℚ(f(z)) over ℚ(f(z)); indeed, f(ω) is often in ℚ(ω, j(ω)), and in such circumstances Weber calls f(ω) a class invariant too. Using such results, Weber computes many class invariants—an end in itself, since the numbers are so beautiful. More recently, results of this type have been applied to determine all the complex quadratic fields with class number 1, and to prove that elliptic curves of certain families always have infinitely many rational points-see [9, 2, 5].

Type
Research Article
Information
Mathematika , Volume 16 , Issue 2 , December 1969 , pp. 283 - 294
Copyright
Copyright © University College London 1969

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