reduction types of elliptic curves 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? 
 A: What you said is almost correct. 
First the list of types is complete if the residue field is algebraically closed. Over a perfect residue field, there are some more types, easy to handle (see nosr's comments). Essentially you identify irreducible components in the special fiber via possible automorphisms of the special fiber. 
Over an imperfect residue field of characteristic 2 or 3 (see again nosr's comment), there are also some more types, decribed in Szydlo's thesis (J. Number Theory, 2004). See also JSE's answer at this question. 
In residue characteristic 2, it is not true that type $I_n^*$ implies potentially multiplicative reduction, but the converse is true (potentially multiplicative reduction $I_m$ implies type $I_n^*$ for some $n$). There is a paper of Dino Lorenzini (in Pure Appli. Maths. Q., special issue in honor of Tate) where among other results, he gives the relation between $n$ and $m$ when the curve has potentially multiplicative reduction $I_n$. The relation involves the different of the minimal extension realizing the good reduction when $K$ is henseilan with algebraically closed $k$. 
The relation between II and II$^{\star}$, when $p\ne 2, 3$, is that II is  obtained by a quadratic twist of II$^\star$, similarly for IV and IV$^\star$ if I remember correctly. You can easily check on some examples using pari/gp.  See Will's comments. 
A: $y^2=x^3-p$ 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 $p\neq 2,3$.
A: The naming comes from elliptic surfaces.
http://en.wikipedia.org/wiki/Elliptic_surface
My recollection, III and III* are quadratic twists $v(\Delta)=3,9$,
as are II and IV* $v(\Delta)=2,8$ and are IV and II* $v(\Delta)=4,10$.
Note that $v(\Delta)=2,3,4$ for II,III,IV, and it goes up/down by 6 when twisting.
I think everything you have said is correct. The idea with "exceptional" good reduction, is that you acquire good reduction after a field extension, unlike [potentially] multiplicative case. The field to do this is easy for $p\ge 5$, but harder for primes above 2,3. See the paper of Kraus (in French, abstract in English). You can work locally, or also get a global field if wanted.
http://link.springer.com/article/10.1007%2FBF02567933
