Three examples of non-commutative boundaries of Shimura varieties

  • Frédéric Paugam
Part of the Aspects of Mathematics book series (ASMA)


Our modest aims in writing this paper were twofold: we first wanted to understand the linear algebra and algebraic group theoretic background of Manin’s real multiplication program proposed in [Man]. Secondly, we wanted to find nice higher dimensional analogs of the non-commutative modular curve studied by Manin and Marcolli in [MM02]. These higher dimensional objects, that we call irrational or non-commutative boundaries of Shimura varieties, are double cosets spaces of the form г/G(ℝ)/P (K), where G is a (connected) reductive Q-algebraic group, P(K) = M(K)ANG(ℝ) is a real parabolic subgroup corresponding to a rational parabolic subgroup PG, and г ⊂ G(ℚ) is an arithmetic subgroup. Along the way, it also seemed clear that the spaces г\G(ℝ)/M(K)A are of great interest, and sometimes more convenient to study. We study in this document three examples of these general spaces. These spaces describe degenerations of complex structures on tori in (multi)foliations.


Modulus Space Conjugacy Class Algebraic Group Elliptic Curf Maximal Torus 
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© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

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  • Frédéric Paugam

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