Degrees of parametrizations of elliptic curves by Shimura curves (Q5949912)

Let \(N\) be a square-free positive integer and \((D,M)\) a pair of positive integers such that \(N=DM\) and the number of prime factors of \(D\) is even. Consider the Shimura curve \(X_0^D(M)\) associated with an Eichler order of level \(M\) in an indefinite quaternion algebra of discriminant \(D\) defined over the field of rational numbers \(\mathbb{Q}\). Let \(J_0^D(M)\) be the Jacobian of \(X_0^D(M)\) and \({\mathfrak I}\) an isogeny class of elliptic curves of conductor \(N\). Then there exists a homomorphism \(\xi_D: J_0^D(M)\to E\) with \(E\) in \({\mathfrak I}\) having the connected kernel which is unique up to multiplication by \(\pm 1\) on \(E\). Note that the Shimura curve \(X_0^1(N)\) is the modular curve \(X_0(N)\) and so \(J_0^1(N)= J_0(N)\). The homomorphism \(\xi_D\) induces the map \(\xi_{D^*}: \Phi_r(J_0^D(M))\to \Phi_r(E)\) on groups of connected components of Néron fibers at \(r\). NEWLINENEWLINENEWLINELet \(\delta^D(M)\) be the degree of the homomorphism \(\xi_D\) and \(c_r\) the number of connected components of the fiber at \(r\) of the Néron model of \textit{E. K. Ribet} and \textit{S. Takahashi} [Proc. Natl. Acad. Sci. USA 94, 11110-11114 (1997; Zbl 0897.11018)] proved that if \(M\) is not a prime, then the following formula is valid: NEWLINE\[NEWLINE\frac{\delta^1(N)}{\delta^D(M)}= \prod_{r|D}c_r.\tag \(*\) NEWLINE\]NEWLINE In this paper the author proves that if \(r\) divides \(D\), then the map \(\xi_{D^*}\) is surjective. Furthermore, the formula \((*)\) is proved for any \(D\) and \(M\). Also, a method of computing \(\delta^D(M)\), when \(D>1\), is presented.