Heights of Heegner cycles and derivatives of L-series

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...

A ug 2 00 1 Derivatives of p - adic L - functions , Heegner cycles and monodromy modules attached to modular forms

Derivatives of Eisenstein series and Faltings heights

In a series of papers, [25], [30], [28], [29], [31], [26], we showed that certain quantities from the arithmetic geometry of Shimura varieties associated to orthogonal groups occur in the Fourier coefficients of the derivative of suitable Siegel-Eisenstein series. It was essential in these examples that this derivative was the second term in the Laurent expansion of a Siegel-Eisenstein series a...

Heights of Heegner Points on Shimura Curves

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. ...

GENERALISED HEEGNER CYCLES AND p-ADIC RANKIN L-SERIES

Introduction 2 1. Preliminaries 6 1.1. Algebraic modular forms 6 1.2. Modular forms over C 9 1.3. p-adic modular forms 11 1.4. Elliptic curves with complex multiplication 12 1.5. Values of modular forms at CM points 14 2. Generalised Heegner cycles 15 2.1. Kuga-Sato varieties 15 2.2. The variety Xr and its cohomology 18 2.3. Definition of the cycles 19 2.4. Relation with Heegner cycles and L-se...