$\newcommand{\Q}{\Bbb Q} \newcommand{\N}{\Bbb N} \newcommand{\R}{\Bbb R} \newcommand{\Z}{\Bbb Z} \newcommand{\C}{\Bbb C} \newcommand{\F}{\Bbb F} $ On page 23 of this paper, one can read:
For example, if $E, E'$ are elliptic curves over $\Q$, then $|N_E(p)-N_{E'}(p)| \leq 1.4 \sqrt p$ for all but finitely many $p$ implies $N_E(p) = N_{E'}(p)$ for all $p$.
Here $N_X(q)$ is the cardinality of $X(\F_q)$ where $X$ is a scheme of finite type over $\Z$ and $q$ is a prime power, and $E,E'$ are supposed to be given by a Weierstrass equation with integer coefficients.
I would like to know why this statement is true, and to have further references proving this fact.
I'm really surprised about the claim above, because it implies in particular the amazing statement
(A) : if $N_E(p) = N_{E'}(p)$ for almost all $p$, then $N_E(p) = N_{E'}(p)$ for all $p$ (even at bad primes).
I think that $N_E(p) = N_{E'}(p)$ implies $N_E(p^2) = N_{E'}(p^2)$, so that $(A)$ might follow from theorem 1 in Strong multiplicity one for the Selberg class (Murty & Murty), by modularity (possibly using this, to relate local $L$-factors to the number of points of $E$ at bad primes – we need to work with Weierstrass models, as mentioned in the comments there). Maybe we can also use Falting's theorem (about Tate's conjecture) together with Chebotarev's density theorem?
But this is anyway very far from the quoted statement above, which only assume a bound $1.4 \sqrt p$ on the difference. According to exericse 3) in §8.4.4 and remark 5) in §6.3.1 of Lectures on $N_X(p)$ by Jean-Pierre Serre, Sato–Tate conjecture implies that $|N_E(p) - N_{E'}(p)|$ is bounded when $p$ runs over the primes (and we could replace $1.4 \sqrt p$ by $1.99\sqrt p$). Still this is not sufficient to apply the theorem 1 cited above.
Related (but different) : (1).