CM points and quaternion algebras
نویسنده
چکیده
2 CM points on quaternion algebras 4 2.1 CM points, special points and reduction maps . . . . . . . . . 4 2.1.1 Quaternion algebras. . . . . . . . . . . . . . . . . . . . 4 2.1.2 Algebraic groups. . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Adelic groups. . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.4 Main objects. . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.5 Galois actions. . . . . . . . . . . . . . . . . . . . . . . 8 2.1.6 Further objects . . . . . . . . . . . . . . . . . . . . . . 8 2.1.7 Measures . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.8 Level structures . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Main theorems: the statements . . . . . . . . . . . . . . . . . 9 2.2.1 Simultaneous reduction maps. . . . . . . . . . . . . . . 9 2.2.2 Main theorem. . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Surjectivity. . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Equidistribution. . . . . . . . . . . . . . . . . . . . . . 12 2.3 Proof of the main theorems: first reductions . . . . . . . . . . 13 2.4 Further reductions . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.1 Existence of a measure and proof of Proposition 2.5 . . 18 2.4.2 A computation. . . . . . . . . . . . . . . . . . . . . . . 19 2.4.3 P -adic uniformization. . . . . . . . . . . . . . . . . . . 22 2.5 Reduction of Proposition 2.13 to Ratner’s theorem . . . . . . 24 2.5.1 Reduction of Proposition 2.21. . . . . . . . . . . . . . . 24 2.5.2 Reduction of Proposition 2.22 . . . . . . . . . . . . . . 25
منابع مشابه
Liftings of Reduction Maps for Quaternion Algebras
We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.
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