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Commutative subrings of certain non-associative rings. (English) Zbl 0990.11018

From the text: Let \(A\) be the ring of integers in an imaginary quadratic field \(k\) of discriminant \(D\). We wish to count the number, denoted \(N(A,R)\), of ways of embedding \(A\) as a commutative subring of the nonassociative ring \(R\). All of the nonassociative rings that we study arise from Coxeter’s order \(R\) in the \(\mathbb{Q}\)-algebra of Cayley’s octonions. Note that by a commutative ring, we mean one that is both commutative and associative. Let \(\varepsilon_A: (\mathbb{Z}/D\mathbb{Z})^\times\to \langle \pm 1\rangle\) be the odd quadratic Dirichlet character associated to \(k\), and let \(L(\varepsilon_A,s)\) be the corresponding Dirichlet \(L\)-function. Then the following result was obtained in [N. Elkies and B. H. Gross, Pac. J. Math., Spec. Issue, 147-158 (1998; Zbl 0981.11036), Theorem 1].
Theorem 1. \[ N(A,R)= \frac{L(\varepsilon_A,-2)} {\zeta(-5)}= -252\cdot L(\varepsilon_A,-2). \] In this paper, the authors give a more streamlined proof using previous results [the authors, Trans. Am. Math. Soc. 351, 1691-1704 (1999; Zbl 0990.11018)].
Theorem 2. \[ N(A,J_2)= \frac{L(\varepsilon_A,-3)} {\zeta(-7)}= 240\cdot L(\varepsilon_A,-3). \] Next, let \(J_2\) be the Abelian group, under matrix addition, of all \(2\times 2\) Hermitian symmetric matrices with entries in \(R\). An element of \(J_2\) has the form \(M= \left(\begin{smallmatrix} A&x\\ \overline{x}&b \end{smallmatrix}\right)\) with \(a,b\in \mathbb{Z}\) and \(x\in R\).

MSC:

11E76 Forms of degree higher than two
11E25 Sums of squares and representations by other particular quadratic forms
17A99 General nonassociative rings
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