Elliptic curves, L-functions and CM-points

CHOW-HEEGNER POINTS ON CM ELLIPTIC CURVES AND VALUES OF p-ADIC L-FUNCTIONS

Introduction 1 1. Basic notions 6 1.1. Motives for rational and homological equivalence 6 1.2. Algebraic Hecke characters 7 1.3. The motive of a Hecke character 8 1.4. Deligne-Scholl motives 9 1.5. Modular parametrisations attached to CM forms 10 1.6. Generalised Heegner cycles and Chow-Heegner points 13 1.7. A special case 15 2. Chow-Heegner points over Cp 15 2.1. The p-adic Abel-Jacobi map 15...

p-ADIC RANKIN L-SERIES AND RATIONAL POINTS ON CM ELLIPTIC CURVES

The aim of this article is to present a new proof of a theorem of Karl Rubin (see [Ru] and Thm. 1 below) relating values of the Katz p-adic L-function of an imaginary quadratic field at certain points outside its range of classical interpolation to the formal group logarithms of rational points on CM elliptic curves. This theorem has been seminal in providing a motivation for Perrin-Riou’s form...

Computing the cardinality of CM elliptic curves using torsion points

Let E/Q be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Ω of the order. If the prime p splits completely in Ω, then we can reduce E modulo one the factors of p and get a curve E defined over Fp. The trace of the Frobenius of E is known up to sign and we need a fast way to find t...

Rankin-selberg L–functions and the Reduction of Cm Elliptic Curves

Let q be a prime and K = Q( √ −D) be an imaginary quadratic field such that q is inert in K. If q is a prime above q in the Hilbert class field of K, there is a reduction map rq : E``(OK) −→ E``(Fq2) from the set of elliptic curves over Q with complex multiplication by the ring of integers OK to the set of supersingular elliptic curves over Fq2 . We prove a uniform asymptotic formula for the nu...

Products of CM elliptic curves