Heights of Heegner Points on Shimura Curves
نویسندگان
چکیده
Introduction 2 1. Shimura curves 6 1.1. Modular interpretation 6 1.2. Integral models 13 1.3. Reductions of models 17 1.4. Hecke correspondences 18 1.5. Order R and its level structure 25 2. Heegner points 29 2.1. CM-points 29 2.2. Formal groups 34 2.3. Endomorphisms 37 2.4. Liftings of distinguished points 40 3. Modular forms and L-functions 43 3.1. Modular forms 44 3.2. Newforms on X 49 3.3. Supercuspidal case 52 3.4. L-functions associated to newforms 54 3.5. Eisenstein series and theta series 57 4. Global intersections 62 4.1. Height pairing 62 4.2. Computing T(m)η 65 4.3. Computing T(m)ξ̂ 66 4.4. Computing T(m)Z 70 4.5. A uniqueness theorem 73 5. Local intersections 75 5.1. Archimedean intersections 75 5.2. Nonarchimedean intersections 79 5.3. Clifford algebras 84
منابع مشابه
Preface Henri Darmon
Modular curves and their close relatives, Shimura curves attached to multiplicative subgroups of quaternion algebras, are equipped with a distinguished collection of points defined over class fields of imaginary quadratic fields and arising from the theory of complex multiplication: the so-called Heegner points. It is customary to use the same term to describe the images of degree zero divisors...
متن کاملEquations of Hyperelliptic Shimura Curves
We describe an algorithm that computes explicit models of hyperelliptic Shimura curves attached to an indefinite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld’s nonarchimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet’s bimodules an...
متن کاملHeegner points on Mumford–Tate curves
1 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 2 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 3 The regulator term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 4 The conjecture . . . . . . . . . . . . . . ....
متن کاملAutomorphisms and Reduction of Heegner Points on Shimura Curves at Cerednik-drinfeld Primes
Let X be a Shimura curve of genus at least 2. Exploiting Čerednik-Drinfeld’s description of the special fiber of X and the specialization of its Heegner points, we show that, under certain technical conditions, the group of automorphisms of X corresponds to its group of Atkin-Lehner involutions.
متن کاملHeegner points, p-adic L-functions, and the Cerednik-Drinfeld uniformization
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 1 Quaternion algebras, upper half planes, and trees . . . . . . . . . . . . . . . . . . . . . . . . . 456 2 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 3 Generalities on Mumford curves . . . . . . . ....
متن کامل