Yokoi’s conjecture.

*(English)*Zbl 1154.11338From the author’s introduction: Let \(p\) be an odd positive integer, write \(d=p^2+4\), and assume that \(d\) is square-free. Let \(K=\mathbb Q(\sqrt{d})\), where \(\mathbb Q\) is the rational field. The author proves the conjecture of H. Yokoi [see Class numbers and fundamental units of algebraic number fields, Proc. Int. Conf., Katata/Jap. 1986, 125–137 (1986; Zbl 0612.12010)] that \(h(d)\) (i.e., the class number of \(K\)) is greater than 1 if \(p>17\). This conjecture is one of the real analogues of the famous problem (solved by Heegner, Stark and Baker) of finding all imaginary quadratic fields with class number 1. Since the fundamental unit of \(K\) is small, it follows from the ineffective theorem of Siegel (similarly to the imaginary case) that there are only finitely many \(p\) for which the special real quadratic field \(K\) has class number 1. So the problem is to find an effective upper bound for \(p\) assuming \(h(d)=1\). Here he proves the following

Theorem. If \(d\) is square-free, \(h(d)=1\) and \(d=p^2+4\) with some odd integer \(n\), then \(d\) 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 the well-known fact that if \(h(d)=1\) then \(d\) is a quadratic nonresidue modulo any prime \(r\) with \(2<r<p\) (Fact B), he obtains the main result:

Corollary. If \(d\) is square-free, and \(d=p^2+4\) with some integer \(p>1861\), then \(h(d)>1\).

It is easy to prove on the basis of the above-mentioned Fact B that \(h(d)>1\) if \(17<p\leq 1861\), so one has a full solution of Yokoi’s conjecture. Note that there are six exceptional fields where \(h(d)=1\), belonging to \(p=1,3,5,7,13,17\).

The same proof with minor modifications works for Chowla’s conjecture, which is a similar class number one problem [see S. Chowla and J. Friedlander, Glasg. Math. J. 17, 47–52 (1976; Zbl 0323.12006)]. He presents that proof in another paper [see Acta Arith. 107, No. 2, 179–194 (2003; Zbl 1154.11339)]. But it seems that the present proof works only for the class number one problem, the class number two problem (for example) remains open. The harder problem of giving an effective lower bound tending to infinity for \(h(p^2+4)\) (the similar statement in the imaginary case was proved by Goldfeld, Gross and Zagier) is also open. We mention above that the fundamental unit is small (hence Siegel’s theorem is applicable), but its logarithm is as large as \(\log p\), so it is large enough to cause a problem if one wants to apply the Goldfeld-Gross-Zagier method.

The starting point of the author’s proof is an idea of J. Beck’s paper [in: Győry, Kálmán (ed.) et al., Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29–August 2, 1996. Berlin: de Gruyter, 55–93 (1998; Zbl 0911.11035)]. In that work Beck excluded some residue classes for \(p\), i.e., he gave effective upper bounds for \(p\) in the class number one case provided \(p\) belongs to certain residue classes. He combined elementary number theory with formulas for special values of zeta-functions related to \(K\) and certain quadratic Dirichlet characters. In this paper, the author uses zeta-functions related to nonquadratic Dirichlet characters; this leads him to elementary algebraic number theory. Using also new elementary ingredients, he is able to exclude all residue classes modulo a given concrete modulus, hence to prove the conjecture.

Theorem. If \(d\) is square-free, \(h(d)=1\) and \(d=p^2+4\) with some odd integer \(n\), then \(d\) 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 the well-known fact that if \(h(d)=1\) then \(d\) is a quadratic nonresidue modulo any prime \(r\) with \(2<r<p\) (Fact B), he obtains the main result:

Corollary. If \(d\) is square-free, and \(d=p^2+4\) with some integer \(p>1861\), then \(h(d)>1\).

It is easy to prove on the basis of the above-mentioned Fact B that \(h(d)>1\) if \(17<p\leq 1861\), so one has a full solution of Yokoi’s conjecture. Note that there are six exceptional fields where \(h(d)=1\), belonging to \(p=1,3,5,7,13,17\).

The same proof with minor modifications works for Chowla’s conjecture, which is a similar class number one problem [see S. Chowla and J. Friedlander, Glasg. Math. J. 17, 47–52 (1976; Zbl 0323.12006)]. He presents that proof in another paper [see Acta Arith. 107, No. 2, 179–194 (2003; Zbl 1154.11339)]. But it seems that the present proof works only for the class number one problem, the class number two problem (for example) remains open. The harder problem of giving an effective lower bound tending to infinity for \(h(p^2+4)\) (the similar statement in the imaginary case was proved by Goldfeld, Gross and Zagier) is also open. We mention above that the fundamental unit is small (hence Siegel’s theorem is applicable), but its logarithm is as large as \(\log p\), so it is large enough to cause a problem if one wants to apply the Goldfeld-Gross-Zagier method.

The starting point of the author’s proof is an idea of J. Beck’s paper [in: Győry, Kálmán (ed.) et al., Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29–August 2, 1996. Berlin: de Gruyter, 55–93 (1998; Zbl 0911.11035)]. In that work Beck excluded some residue classes for \(p\), i.e., he gave effective upper bounds for \(p\) in the class number one case provided \(p\) belongs to certain residue classes. He combined elementary number theory with formulas for special values of zeta-functions related to \(K\) and certain quadratic Dirichlet characters. In this paper, the author uses zeta-functions related to nonquadratic Dirichlet characters; this leads him to elementary algebraic number theory. Using also new elementary ingredients, he is able to exclude all residue classes modulo a given concrete modulus, hence to prove the conjecture.

Reviewer: Olaf Ninnemann (Berlin)