TITLE: Does p-adic $L$- function determine the $L$ function
QUESTION [20 upvotes]: Let $E_1$  and $E_2$ be two Elliptic curve defined over $\mathbb Q$ . Let $p$ be an fixed given  odd prime of $\mathbb Q$ at which both the curves have good ordinary reduction. Moreover  p-adic $L$-function  of $E_1$ and $E_2$ are same . Does it mean that  the complex $L$-function of $E_1$ and $E_2$  are also same ?
Is there some sufficient criteria on p-adic $L$-functions such that such that the $L$ function is determined?

REPLY [4 votes]: Yes, it is true that the $p$-adic $L$-function of an elliptic curve $E$ over $\mathbf{Q}$ determines the isogeny class of $E$. As mentioned in the Luo-Ramakrishnan article pointed out by Idoneal, this follows from analytical results of Rohrlich. I thought I would sketch the argument of Rohrlich because it's interesting.
So, let us assume $L_p(E_1,\cdot)=L_p(E_2,\cdot)$ for two elliptic curves $E_1,E_2$ with good ordinary reduction at $p$. By the interpolation property of the $p$-adic $L$-function, this means that
\begin{equation}
\frac{L(E_1,\chi,1)}{\Omega_{E_1}^\pm} = \frac{L(E_2,\chi,1)}{\Omega_{E_2}^\pm}
\end{equation}
for all Dirichlet characters $\chi$ of $p$-power conductor. Here $\pm = \chi(-1)$ and $\Omega^+$ (resp. $\Omega^{-}$) is the real (resp. imaginary) period of a Néron differential.
Now there is this nice analytical result by Rohrlich (On $L$-functions of elliptic curves and cyclotomic towers, Invent. Mat. 75, 1984) : if $\chi$ is a Dirichlet character of conductor $p^m$, then the average value of $L(E,\chi^{\sigma},1)$, where $\chi^\sigma$ runs through the conjugate characters of $\chi$, tends to $1$ as $m \to +\infty$. This is proved by writing $L(E,\chi,1)$ as a quickly converging series and using clever analytical arguments.
From this we deduce that $\Omega_{E_1}^{\pm}=\Omega_{E_2}^{\pm}$. This implies that there is an isogeny $\varphi : E_1 \to E_2$ (which is a priori defined over $\mathbf{C}$) such that $\varphi^* \omega_{E_2}$ is a non-zero rational multiple of $\omega_{E_1}$. By taking the sums of the conjugates of $\varphi$, we deduce that $E_1$ and $E_2$ are isogenous over $\mathbf{Q}$ (and thus their complex $L$-functions coincide).
Remark : we cannot always deduce that $E_1$ and $E_2$ are isomorphic over $\mathbf{Q}$. For example $E_1=15a1$ and $E_2=15a4$ are $2$-isogenous and have the same Néron periods $\Omega_E^{\pm}$, so they have the same $p$-adic $L$-function, but they are not isomorphic. To me this rather indicates that the definition of $p$-adic $L$-function is not the right one (as pointed out by Olivier in his comment). If we want a theory of $p$-adic $L$-functions which looks like as much as possible as the theory of complex $L$-functions, then the $p$-adic $L$-function should be invariant under isogeny.