The formal completion of the Néron model of \(J_ 0(p)\) (Q1182647)

Let \(p>3\) be prime. Let \(J_ 0(p)\) be the Jacobian of the modular curve \(X_ 0(p)\) and let \({\mathcal J}_{\mid\mathbb{Z}}\) be the Néron model of \(J_ 0(p)\). Let \(U_ p\), \(T_ \ell\in M_ g(\mathbb{Z})\) be the matrices of the Atkin-Lehner operator and the Hecke operators acting on \(S_ 2(\Gamma_ 0(p),\mathbb{Z})\) for all primes \(\ell\neq p\). Form the formal power series \(\sum^ \infty_{n=1}A_ n\cdot n^{-s}=(I_ g-U_ p\cdot p^{-s})^{-1}\cdot\prod_ \ell(I_ g-T_ \ell\cdot p^{- s}+I_ g\cdot p^{1-2s})^{-1}\). Let \(L(X,Y)\) be the \(g\)-dimensional formal group law with logarithm \(f(x)=\sum^ \infty_{n=1}{1\over n}A_ nX^ n\in\mathbb{Q}[X_ 1,\ldots,X_ g]^ g\). The author then establishes the following theorem: The group law \(L(X,Y)\) is defined over \(\mathbb{Z}\) and is isomorphic to the formal completion of \({\mathcal J}\) along the zero section. The proof is based on certain results of B. Mazur. --- Similar objects to the above exist in the theory of rank-2-Drinfeld modules and one wonders if an analogous result may be established.