# good/bad reduction modulo $p$ in Rational Points on Elliptic Curves vs. Arithmetic of Elliptic Curves

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?

• 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. – reuns Apr 19 at 3:38
• $\Bbb F_p$ 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 $R=\Bbb Z$. – Lord Shark the Unknown Apr 19 at 3:41
• @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? – quietkid Apr 19 at 3:55
• @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. – quietkid Apr 19 at 3:57