Hida families and rational points on elliptic curves

Abstract

Let E be an elliptic curve over Q of conductor N = Mp, having a prime p‖N of multiplicative reduction. Because E is modular, it corresponds to a normalised weight two eigenform on Γ0(N), whose q-expansion is denoted f = ∑n anq . LetX := hom(Z×p ,Z×p ) Z/(p−1)Z×Zp, which contains Z as a dense subset by associating to k ∈ Z the character x → xk−2. Denote by A(U) the ring of Cp-valued p-adic analytic functions on a compact open subset U of X. Hida’s theory associates to f a neighborhood U of 2 ∈ X (which can be assumed, for simplicity, to be contained in the residue class of 2 modulo p − 1) and a formal q-expansion