0

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?

quietkid
  • 151
  • 1
    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 '19 at 03:38
  • 1
    $\Bbb F_p$ 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 $R=\Bbb Z$. – Angina Seng Apr 19 '19 at 03:41
  • @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 '19 at 03:55
  • @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 '19 at 03:57

0 Answers0