Let $E/K$ be an elliptic curve, where $K$ is a complete local field with residue field $k$ and char$(k) = p$. I'm trying to make sense of Kodaira symbols and Tate's algorithm.
My current understanding is:
I$_0$ = good reduction
I$_n$ = multiplicative reduction with $\nu(j) = -n$.
I$_n^\ast$ = potential multiplicative reduction, eventually becoming I$_n$ in a field extension (so $\nu(j) = -n$).
I$_0^\ast$ = "non-exceptional" potential good reduction.
II, II$^\ast$, III, III$^\ast$, IV, IV$^\ast$ = "exceptional" potential good reduction. These can only happen when the $j$-invariant is equivalent to 0 or 1728 modulo $p$, or when $p = 2, 3$ (where everything is more complicated…)
Is this correct? Silverman's Advanced Topics in the Arithmetic of Elliptic Curves has a good table of reduction types when $k$ is algebraically closed, but I haven't been able to find something analogous for more general fields which gives me an overview of the possibilities. Also, why are Kodaira symbols named the way they are? For example, how are the reduction types II and II$^\ast$ related?
Best Answer
$y^2=x^3-p$ has reduction type $II$, $y^2=x^3-1/p$ has reduction type $II^*$.
$y^2=x^3-p^2$ has reduction type $IV$, $y^2=x^3-1/p^2$ has reduction type $IV^*$.
Moreover, these examples are universal, in that everything of those fiber types looks like those equations up to linear change of variables and higher-order terms.
I believe this is the source of the names.
Similarly, $y^2=x^3-px$ is $III$, and $y^2=x^3-x/p$ is $III^*$.
This is only for $p\neq 2,3$.