A note on quadratic fields in which a fixed prime number splits completetly. III
For a fixed prime number \(p\), denote by \(M(p)^+\) the set of all real quadratic fields in which \(p\) splits, and for a quadratic field \(K\) in \(M(p)^+\) denote by \(\delta_p^+(K)\) the order of the ideal class of \(K\) containing a prime ideal of \(K\) over \(p\). Then the author is concerned with the image of the map \(\delta_p^+: M(p)^+\to N\), \(K\to \delta_p^+(K)\). In Part II of his series [Proc. Japan Acad., Ser. A 75, 150-151 (1999; Zbl 0997.11088)], he showed that the image of \(\delta_p^+\) contains \(2n\) for all \(n\geq 0\) and any \(p\). In this paper, he shows the following: Under the assumption of the \(abc\) conjecture [cf. \textit{P. Vojta}, Diophantine approximations and value distribution theory, Chap. 5, Lecture Notes Math. 1239, Springer (1987; Zbl 0609.14011)], (i) the complement \(N\setminus \operatorname {Im} \delta_p^+\) is a finite set for any prime \(p\), (ii) \(\operatorname {Im} \delta_p^+\) coincides with \(N\) for infinitely many \(p\).
scientific article; zbMATH DE number 1443394