Congruences of Hurwitz class numbers on square classes
type: Item
instance of: scholarly article
title: Congruences of Hurwitz class numbers on square classes
author: Olivia Beckwith
author: Olav K. Richter
author: Martin Raum
published in: Advances in Mathematics
publication date: 2022-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/2203.11273
review text: 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 arithmeti…
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 7605920
zbMATH Keywords: Hurwitz class numbers
zbMATH Keywords: Ramanujan-type congruences
zbMATH Keywords: holomorphic projection
MaRDI profile type: Publication
OpenAlex ID: W4294837467
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DOI: 10.1016/J.AIM.2022.108663
description: scientific article; zbMATH DE number 7605920