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On rigid tuples in linear groups of odd dimension. (English) Zbl 0945.12001

This paper is about “rigid local systems” (for an introduction to those systems, see N. Katz’s book “Rigid local systems” [Ann. Math. Stud. 139 (1996; Zbl 0864.14013)]). One says that a tuple \((g_1,\dots,g_r)\) of elements of \(GL_n (\mathbb{C})\), such that \(g_1,\dots, g_r\in \mathbb{C}^*\), is linearly rigid if, for any other tuple \((h_1,\dots, h_r)\) in \(GL_n(\mathbb{C})\), such that \(h_i\) is conjugate to \(g_i\) \((1\leq i\leq r)\) and \(h_1\cdots h_r= g_1\cdots g_r\), then there exists an \(x\) in \(GL_n(\mathbb{C})\) such that \(h_i=xg_i x^{-1}\) for all \(i\) (this notion is introduced by K. Strambach and H. Völklein in J. Reine Angew. Math. 510, 57-62 (1999; Zbl 0931.12006)). The authors give new rigid tuples in \(GL_n(\mathbb{C})\). Applications to Galois relizations over \(\mathbb{Q}\) are given, for the groups \(PGL_{2m+1} (\mathbb{F}_q)\), resp. \(PU_{2m+1} (\mathbb{F}_{q^2})\) \((q=p^l\) with \(p\) an odd prime number, \(q\neq 3\), \(m>\phi (q-1)\), resp. \(m> \phi(q+1))\).

MSC:

12F12 Inverse Galois theory
14H30 Coverings of curves, fundamental group
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References:

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