I stumbled upon this question, and I can't think of how to do it, or what kind of results to use. The question is as follows:
Let $$y^2=x^3+ax+b$$ be an elliptic curve ($a,b$ integers), and let $p \geq 5$ be a prime with good reduction. Then I have to show that this curve has infinitely many solutions in $\mathbb{Z}_p$, but I cant think of what results to use to try and prove this, I'm guessing something like Hensel's Lemma might work, but I can't see how. So I was wondering if I could get some hints on this.
Thank you
Here is a solution that works for $p\geq 11$. You will need to modify the proof to make it work for $p=5$ and $p=7$.
Since $p\geq 11$, and $p$ is a prime of good reduction, the Hasse bounds tell us that $$|E(\mathbb{F}_p)|\geq p+1-2\sqrt{p}\geq 5.$$ Since there are at most $3$ solutions modulo $p$ with $y\equiv 0 \bmod p$, it follows that there is at least one point in $E(\mathbb{F}_p)$ with non-zero $y$ coordinate.
Let $(\alpha,\beta)$ be a point on $E(\mathbb{F}_p)$ with $\beta\not\equiv 0 \bmod p$, and let $\gamma\in \mathbb{Z}_p$ be any $p$-adic integer such that $\gamma\equiv \alpha \bmod p$. Put $f(x)=x^3+ax+b$, and consider the polynomial $q(y)=y^2-f(\gamma)$. Then any $\delta_0\equiv \beta \bmod p$ satisfies $q(\delta_0)=\delta_0^2-f(\gamma)\equiv 0 \bmod p$, and $q'(\delta_0)=2\delta_0\equiv 2\beta\not\equiv 0 \bmod p$. Hence, by Hensel's Lemma $q(y)$ has a root $\delta$ in $\mathbb{Z}_p$, and so $(\gamma,\delta)\in E(\mathbb{Z}_p)$. Since the choice of $\gamma$ was arbitrary among all elements of the form $\alpha+p\mathbb{Z}_p$, it follows that $E(\mathbb{Z}_p)$ is infinite.
PS. One can also do this using formal groups. For instance, Proposition 6.3 of Chapter VII of Silverman's "The Arithmetic of Elliptic Curves" implies that $E(\mathbb{Z}_p)$ is infinite.