On Néron Models, Divisors and Modular Curves

For p a prime number, let X0(p)Q be the modular curve, over Q, parametrizing isogenies of degree p between elliptic curves, and let J0(p)Q be its jacobian variety. Let f : X0(p)Q → J0(p)Q be the morphism of varieties over Q that sends a point P to the class of the divisor P − ∞, where ∞ is the Q-valued point of X0(p)Q that corresponds to PQ with 0 and ∞ identified, equipped with the subgroup scheme μp,Q. We suppose that X0(p)Q has genus at least one, i.e., that p = 11 or p > 13. Then f is a closed immersion. Let 0 denote the other cusp of X0(p)Q. The Q-valued point f(0) of J0(p)Q is well known to be of order n, the numerator of (p− 1)/12, expressed in lowest terms. Let now X0(p) denote the model over Z of X0(p)Q as described by [8] and [15]: it is the compactified coarse moduli space for generalized elliptic curves with finite locally free subgroup schemes of rank p that meet all irreducible components of all geometric fibres. Let X0(p) denote the minimal regular model of X0(p), and let J0(p) be the Néron model over Z of J0(p)Q. See [8] for a description of the semi-stable curve X0(p). By the defining properties of J0(p), the morphism f extends uniquely to a morphism f : X0(p) → J0(p), where X0(p) is the open part of X0(p) where the morphism to Spec(Z) is smooth; X0(p) is the complement of the set of double points in the fibre X0(p)p over Fp. Robert Coleman asked about how the image of X0(p) p under f intersects C, the closure in J0(p) of the group generated by f(0). We will prove, see Theorem 8.2, for all p for which X0(p)Q has genus at least two, that the intersection consists just of the two obvious elements f(0)p and f(∞)p. (Of course, when X0(p)Q has genus one, f is an isomorphism, hence the intersection is all of Cp.) This result is used by Coleman, Kaskel and Ribet in [4] to study the inverse image under f of the torsion subgroup of J0(p)(C).