MAPS ON GROUPS OF CONNECTED COMPONENTS INDUCED FROM PARAMETRIZATIONS OF ELLIPTIC CURVES BY SHIMURA CURVES

S. Takahashi (USA)

Received December 17, 2007; Revised December 26, 2008

Abstract

An optimal parametrization of an elliptic curve by a Shimura curve induces a map on the groups of connected components of  reductions of Néron models of Jacobians of the Shimura curve and the elliptic curve where p is a prime number dividing the discriminant of the Shimura curve. It is known that for every prime  if the Galois representation on the group of -division points of the elliptic curve is irreducible, then  does not divide the order of the cokernel of the map on the groups of connected components. It is believed that the statement is true without the irreducibility condition on the Galois representation and hence that the map on the groups of connected components is surjective. In this paper, we will prove a similar statement replacing the irreducibility condition with the condition that  does not divide the order of the group of roots of unity in the multiplicative group of p-adic numbers.

Keywords and phrases: elliptic curves, Shimura curves, groups of connected components.