Congruences of Hurwitz class numbers on square classes
In this paper, the authors study the divisibility of Hurwitz class numbers on certain arithmetic progressions and congruences for class numbers of imaginary quadratic fields whose discriminants vary within a given square class. For this discussion, let \(H(D)\) denote the Hurwitz class number, let \(\ell\) be a prime exceeding 3, and let \(a\) and \(b\) be integers with \(a>0\). When \(-b\) is a square modulo \(a\), the authors refer to congruences of the type \(H(an+b)\equiv 0\pmod\ell\) as non-holomorphic Ramanujan-type congruences due to the fact that the generating series \(H(an+b)\) has a non-holomorphic modular completion in that case. In a previous paper [Proc. Natl. Acad. Sci. U.S.A. 117, No. 36, 21953--21961 (2020; Zbl 1485.11088)], the authors used a holomorphic projection argument to prove that when such a non-holomorphic congruence holds, the divisibility \(\ell \mid a\) holds. In the first main theorem in the present paper, a holomorphic projection argument is again used in order to prove that under these conditions the divisibility \(\ell \mid b\) also holds. A second main theorem, in which it is not assumed that \(-b\) is a square modulo \(a\), states that if \(H(an+b)\equiv 0\pmod\ell\) holds for all integers \(n\), then \(H(an+bu^2)\equiv 0\pmod\ell\) holds for all integers \(n\) and \(u\) with \(\mathrm{g.c.d.}(u,a)=1\). In the case that \(-b\) is not a square modulo \(a\), the congruences \(H(an+b)\equiv 0\pmod\ell\) can be studied via the classical theory of modular forms, and the proof in this case parallels previous work by \textit{C.-S. Radu} on partition congruences [J. Reine Angew. Math. 672, 161--175 (2012; Zbl 1276.11165); Trans. Am. Math. Soc. 365, No. 9, 4881--4894 (2013; Zbl 1336.11071)]. Two applications of this second theorem are presented. The first gives upper bounds for \(\mathrm{ord}_p(a/\mathrm{g.c.d.}(a,b))\), for primes \(p\), for a maximal Ramanujan-type congruence modulo \(\ell\) for the Hurwitz class numbers on the arithmetic progression \(a\mathbb Z+b\). Here maximal means that there is no Ramanujan-type congruence modulo \(\ell\) on any arithmetic progression \(a'\mathbb Z+b'\) that is properly contained in \(a\mathbb Z+b\). The second application describes a relationship between Ramanujan-type congruences for Hurwitz class numbers and congruences for class numbers of imaginary quadratic fields whose discriminants vary in a square class modulo \(a\).
scientific article; zbMATH DE number 7605920
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