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Generalized automorphs of the sum of four squares. (English) Zbl 0232.10031

Denote by \(H\) the Hamiltonian quaternion algebra and by \(L\) the Lipschitz ring of integral quaternions. Let \(x = x_0 + x_1i +x_2j + x_3k\) and \(y = y_0 + y_1i +y_2j + y_3k\), \(\bar x\) the conjugate of \(x\). If \(T = (t_{\alpha\beta}\) is a \(4\times 4\) rational matrix, define the linear map \(\tau\colon H\to H\) by \(\tau x =y\), where \(y_\alpha = \sum_{t_{\alpha\beta}} x_\beta\) \((\alpha = 0,1,2,3)\). If \(T\) is integral and satisfies the equation \(T'T = mI\), where \(T'\) denotes the transpose of \(T\) and \(m\) is a positive integer, then the author says that \(T\) (and also \(\tau\)) is a generalized automorph of the sum of four squares. The main result is the theorem:
Let \(\tau\) be a generalized automorph of the sum of four squares. Then, there exist a primitive quaternion \(a\) of odd norm, a quaternion \(b\) in \(L\) and an automorphism \(\mu\) of \(L\), such that \[ \tau x = a\mu(x)b\quad\text{if }\det \tau > 0,\qquad \tau x = a\mu(\bar x) b\quad\text{if }\det \tau < 0. \] Moreover, if \(a'\), \(b'\) and \(\mu'\) also satisfy the above conditions, then there is a unit \(h\) of \(L\) such that \(a' = ah^{-1}\), \(b'=hb\) and \(\mu'(x) =h\mu(x)h^{-1}\).

MSC:

11R52 Quaternion and other division algebras: arithmetic, zeta functions
11E25 Sums of squares and representations by other particular quadratic forms
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