Chowla’s conjecture. (English) Zbl 1154.11339

From the author’s introduction: Let \(K=\mathbb Q(\sqrt(d))\), where \(\mathbb Q\) is the rational field and \(d\) is a square-free positive integer, and let \(h(d)\) be the class number of this field. In [S. Chowla and J. Friedlander, Glasg. Math. J. 17, 47–52 (1976; Zbl 0323.12006)], S. Chowla conjectured that \(h(4p^2+1)>1\) if \(p>13\) is an integer, which is proved to be true in this paper. The work here has its origins in the author’s paper [Acta Arith. 106, No. 1, 85–104 (2003; Zbl 1154.11338)], in which he established a conjecture of H. Yokoi that \(h(p^2+4)>1\) for \(p>17\). In fact, essentially the same proof works with appropriate modifications. Note that Siegel’s theorem tells us that the class number is greater than 1 once \(p\) is sufficiently large, in both cases; however, Siegel’s theorem does not indicate what “sufficiently large” means. Here the author determines that using a quite different method. His main result is as follows.
Theorem. If \(d\) is square-free, \(h(d)=1\) and \(d=4p^2+1\) with some positive integer \(p\), then \(p\) is a square for at least one of the following moduli: \(q=5,7,41,61,1861\) (that is, \((d/q)=0\) or 1 for at least one of the listed values of \(q\)).
Combining this with Fact B (which implies that if \(h(d)=1\), then \(d\) is a quadratic nonresidue modulo any prime \(r\) with \(2<r<p\)) he obtains:
Corollary. If \(d\) is square-free, and \(d=4p^2+1\) with some integer \(p>1861\), then \(h(d)>1\).
What concerns the small solutions, in the same way as in the author’s cited paper, he can easily prove (see Section 2) that \(h(4p^2+1)>1\) if \(13<p\leq 1861\). Hence Chowla’s conjecture follows. He searches these final few cases to show that \(h(4p^2+1)=1\) only for \(p=1,2,3,5,7,13\).
The main lines of the proof are the same as in the author’s cited paper, but some modifications are needed; the most significant modifications can be found in the statement and proof of Lemma 1. The present proof also requires computer work.


11R11 Quadratic extensions
11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
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