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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?

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The formation of the Neron model over henselian discrete valuation rings commutes with scalar extension to the maximal unramified extension and its completion, so for any "table" of reduction types it is often sufficient to consider only separably closed residue fields. Hence, to the extent the residue field is perfect, it usually may as well be algebraically closed. (It isn't clear if Tate's algorithm works for imperfect residue field $k$ of char. 2 or 3, due to the existence of non-smooth regular Weierstrass cubics over such $k$.) For much more, read 10.2 in Qing Liu's textbook. –  user28172 Apr 18 '13 at 5:45
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3 Answers

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.

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I believe this is incorrect. $II$ is a quadratic twist of $IV*$, and and $IV$ is a quadratic twist of $II*$. –  Will Sawin Apr 24 '13 at 15:56
Thanks Will for the correction ! And III$^\star$ is a quadratic twist of III. –  Qing Liu Apr 24 '13 at 17:11
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$y^2=x^3-p$ 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 $p\neq 2,3$.

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Thanks Will! Could you point me towards a reference for those examples being universal? –  stl Apr 28 '13 at 14:58
Unfortunately I don't have a reference. But the idea is simple - just put the elliptic curve in Weierstrass form $y^2=x^3-g_2x-g_3$. Then scale $g_2$ and $g_3$ so that one coefficient is as appropriate - I mean that if the fiber type is $II$, scale $g_3$ until it's $p$, but if it's $III^*$, scale $g_2$ until it's $1/p$. By Tate's algorthim, it's clear we can do this without changing the isomorphism class of the elliptic curve over the maximal unramified extension of $\mathbb Q_p$. –  Will Sawin Apr 28 '13 at 20:09
Then the other one of $g_2$ and $g_3$ must make a higher-order contribution to the discriminant. This is what I mean by higher-order terms. This is easy to check from the $j$-invariant: If $g_2^3/(g_2^3-27g_3^2)$ is $0$ at $p$, then $g_2^3$ must vanish to higher order at $p$ than $g_3^2$. (Again, $p\neq 2,3$.) –  Will Sawin Apr 28 '13 at 20:11
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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.


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