Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of 𝑝-adic 𝐿-Functions

Abstract

Let Γ \Gamma be a congruence subgroup of type ( N 1 , N 2 ) (N_1,N_2) and of level N N . We study congruences between weight 2 normalized newforms f f and Eisenstein series E E on Γ \Gamma modulo a prime p \mathfrak {p} above a rational prime p p . Assume that p ∤ 6 N p\nmid 6N , E E is a common eigenfunction for all Hecke operators and f f is ordinary at p \mathfrak {p} . We show that the abelian variety associated to f f and the cuspidal subgroup associated to E E intersect non-trivially in their p p -torsion points. Let A A be a modular elliptic curve over Q \mathbb {Q} with good ordinary reduction at p p . We apply the above result to show that an isogeny of degree divisible by p p from the optimal curve A 1 A_1 in the Q \mathbb {Q} -isogeny class of elliptic curves containing A A to A A extends to an étale morphism of Néron models over Z p \mathbb {Z}_p if p > 7 p>7 . We use this to show that p p -adic distributions associated to the p p -adic L L -functions of A A are Z p \mathbb {Z}_p -valued.