Deforming the Gauss-Manin Connection

§0. The Gauss-Manin Connection. Let p > 2 be a prime and N ≥ 4. The p-adic modular curve X1(Np) will be denoted X. We let X = W0 ∪W∞ denote the usual decomposition of X as the union of two wide open sets. Hence Wi (i = 0 or ∞) is a wide open neighborhood of the ordinary component Zi containing the cusp i (= 0 or ∞) and the intersection W = W∞ ∩ W0 is the union the supersingular annuli. We have Z∞ := W∞ \ W and Z0 := W0 \ W . Let X(v) be the largest connected wide open neighborhood of Z∞ on which the canonical subgroup is defined. Let π : E −→ X denote the universal generalized elliptic curve over X and let H := H DR(E/X) denote the De Rham cohomology sheaf on X. Let ∇ : H −→ H denote the Gauss-Manin connection on H. One knows from Katz that H has a natural decomposition as H = ω−1 ⊕ ω