Abstract: Fix a square free positive integer N, and an isogeny class of elliptic curves of conductor N over Q. Consider a decomposition N = DM. Let J be the Jacobian of Shimura curve associated with the Eichler order of level M in the indefinite quaternion algebra of discriminant D over Q. Now by the works of Wiles, Taylor-Wiles, and Jacquet-Langlands, there is an E in the isogeny class and a parametrization J --> E having the connected kernel.
In this talk, I study the map on the groups of the connected components of the Neron fibers at p induced from J --> E where p is a prime dividing N. I describe the order of the image of the map in terms of the character group of the toric part of the Neron fiber at p of J and the action of Hecke operators on it. Combining this description with a result of Bertolini-Darmon, I show that if p divides D, then the map is surjective. As an application, I describe a formula for the degree of J --> E which enables us to compute the degree in the setting of linear algebra. Moreover, I talk about some relations among degrees when D and M vary, which sharpen the result of Ribet and myself.