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On the 2-part of the class number of imaginary quadratic number fields. (English) Zbl 0329.12003

This paper uses a “type number formula” from the theory of quaternion algebras to obtain information on the 2-part of the class number of imaginary quadratic number fields. The type number of a positive definite quaternion algebra \(\mathfrak A\) over \(\mathbb Q\) is just the number of isomorphism classes of certain kinds of orders (in this case Eichler orders) in \(\mathfrak A\). Denote by \(h(-m)\) the class number of \(\mathbb Q(\sqrt{-m})\), \(m\) a square free positive integer. The type number formula involves terms of the form \(2^{-r}h(-m)\) for some \(m\) and some positive integer \(r\). The results in the paper follow from the observation that the type number being an integer imposes certain relations on the class numbers that appear in the formula. Examples of the kind of results obtained are: Let \(p\) be a prime number. \[ \text{If } p\equiv 1(8), \text{then } h(-p) + h(-2p)\equiv \begin{cases} 0(8)\text{ if } p\equiv 1(16) \\ 4(8)\text{ if } p\equiv 9(16). \end{cases} \tag{a} \] \[ \text{If } p\equiv 7(8), \text{then } h(-2p) \equiv \begin{cases} 0(8)\text{ if } p\equiv 15(16) \\ 4(8)\text{ if } p\equiv 7(16). \end{cases} \tag{b} \]
Equation (a) is related to results of Hasse and Barrucand-Cohn. Similar results hold for \(p\equiv 3\text{ or } 5(8)\). Let \(p\) and \(q\) be distinct primes \(\ge 3\). \[ \text{If } p\equiv 1(8), q\equiv 1(8) \text{ and }(p/q ) = -1, \text{then } \tag{c} \] \[ h(-pq) + h(-2pq) \equiv \begin{cases} 0(8)\text{ if } pq\equiv 1(16) \\ 8(16 )\text{ if } pq\equiv 9(16). \end{cases} \] \[ \text{If } p\equiv 3(8), q\equiv 5(8) \text{ and }(p/q ) = +1, \text{then } \tag{d} \] \[ 4h(-p) + 2h(-2q) + h(-2pq) \equiv \begin{cases} 4(16)\text{ if } p\equiv 11(16) \text{ or } p=3 \\ 12(16 )\text{ if } p\equiv 3(16), \quad p\ne 3. \end{cases} \]
Relations of a similar kind are given for all \(h(-m)\) where \(m\) has 3 or fewer prime factors (except in the case \(m = pqr\) with \(m\equiv 3(8))\).
The method can be used to obtain information about \(h(-m)\) for any \(m\), but the statement of the results and the proofs become (much) messier as the number of primes dividing \(m\) increases.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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[1] Barrucand, P; Cohn, H, Notes on primes of type x2 + 32y2, class number and residuacity, J. reine angew. math., 238, (1969)
[2] Bauer, H, Zur berechnung der 2-klassenzahl der quadratischen zahlkörper mit genau zwei verschiedenen diskriminantenprimteilern, J. reine angew. math., 238, 42-46, (1971) · Zbl 0215.07202
[3] Deuring, M, Algebre, (1935), Springer-Verlag New York
[4] Eichler, M, Zur zahlentheorie der quaternionen algebren, J. reine angew. math., 195, 127-151, (1955) · Zbl 0068.03303
[5] Hasse, H, Über der klassenzahl des Körpers \(P(−2p)\^{}\{12\}\) mit einer primzahl p ≠ 2, J. number theory, 1, 231-234, (1969) · Zbl 0167.32302
[6] Hasse, H, Über die teilbarkeit durch 2^{3} der klassenzahl imaginärquadratischer zahlkörper mit genau zwei verschiedenen diskriminantenprimteilern, J. reine angew. math., 241, (1970) · Zbl 0212.08002
[7] Pizer, A, Type numbers of eichler orders, J. reine angew. math., 264, (1973) · Zbl 0274.12008
[8] Rédei, L; Reichardt, H, Die anzahl der durch 4 teilbaren invarianten der klassengruppe eines beliebigen quadratischen zahlkörpers, J. reine angew. math., 170, 69-74, (1934) · Zbl 0007.39602
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