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Rational points and automorphic forms. (English) Zbl 1096.11022

Hida, Haruzo (ed.) et al., Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press (ISBN 0-8018-7860-8/hbk). 733-742 (2004).
Let \(G\) be an \(F\)-anisotropic inner form of a split semi-simple group \(\widetilde G\) of adjoint type over a number field \(F\). Let \(\rho_F: G\to D^x\) be an \(F\)-group morphism from \(G\) to the multiplicative group of a central simple algebra \(D\) over \(F\). Let \(N(\rho,B)=\# \{\gamma\in G(F)\); \(H(\rho_F(\gamma))\leq B\}\), where \(H\) denotes a global height function on \(D\). Let \(\rho_\kappa\) denotes the irreducible algebraic representation of \(\widetilde G\) associated with a dominant weight \(\kappa=\sum_{\alpha\in\Phi^+}\alpha+\sum_{1\leq i\leq r} \alpha_i\). Here a Borel subgroup with maximal split torus in \(\widetilde G\) is fixed and \(\Phi\) denotes the root system, and \(\{\alpha_1,\dots, \alpha_r\}\) is the set of simple roots. The authors give an outline of their proof of the main result, which says that \(N(\rho_\kappa,B)=(c_\kappa/ (r-1)!)B(\log B)^{r-1} (1+o (1))\), where \(c_\kappa=\lim_{s\to 1}(s-1)^r \int_{G({\mathbf A})}H(\rho_F (g))^{-s}dg\) with \(dg\) a suitably normalized Haar measure on \(G({\mathbf A})\).
For the entire collection see [Zbl 1051.11005].

MSC:

11G50 Heights
22E50 Representations of Lie and linear algebraic groups over local fields
11F11 Holomorphic modular forms of integral weight
14G05 Rational points
11G35 Varieties over global fields
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