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

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