On the 2-part of the class number of imaginary quadratic number fields (Q1227024)

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.