## 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.

### MSC:

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

### Citations:

Zbl 0323.12006; Zbl 1154.11338
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