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I am trying to understand what an elliptic curve mod $\pi$ vs mod $p$ is. Basically I am confused about the treatment given in Silverman's two books.

The definition for mod $p$ in Rational Points of Elliptic Curves is basically sending the group of points $C(\mathbb{Q})$ to $C(\mathbb{F}_p)$. From my understanding, $C$ has a good reduction if $C/\mathbb{F}_p$ is not singular, i.e. $p$ does not divide the discriminant of the new equation.

But in Arithmetic of Elliptic Curves they introduce a lot of other propositions such as "Let $E/K$ be an elliptic curve. Then E has potential good reduction if and only if its $j$-invariant is integral, i.e., if and only if $j(E) ∈ R$". ($R$ being the ring of integers of $K$).

I was really confused because I can't understand what $R$ would be for the field $\mathbb{F}_p$ until I read the definition of algebraic number field. So I don't think $\mathbb{F}_p$ is a number field...so my question is what is going on with these two definitions? Are they just different levels of complexity or is it just something completely different about local fields?

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    $\begingroup$ Do you know $\Bbb{Q}_p$ the $p$-adic integers ? Sending $E/\Bbb{Q}$ to $E/\Bbb{Q}_p$ is the main step in Silverman, because then reducing $E/\Bbb{Q}_p \to E/\Bbb{F}_p$ is natural and easy. $\endgroup$ – reuns Apr 19 at 3:38
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    $\begingroup$ $\Bbb F_p$ is a finite field, not a number field. A number field is a finite extension of $\Bbb Q$. In the case where $K=\Bbb Q$ itself, then $R=\Bbb Z$. $\endgroup$ – Lord Shark the Unknown Apr 19 at 3:41
  • $\begingroup$ @reuns ah i see! i haven't learned p-adic integers yet. so basically it's the same map but there's an additional step? $\endgroup$ – quietkid Apr 19 at 3:55
  • $\begingroup$ @LordSharktheUnknown Yes I can see that $\mathbb{F}_p$ is not a number field. My question was why is there not the immediate mention of $\mathbb{F}_p$ in the second book. As in where are all these other conditions coming from if it's the same thing as the first map. $\endgroup$ – quietkid Apr 19 at 3:57

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