In this post, we classify one-dimensional algebraic groups over an algebraically closed field. This post was later extended in this one.
Motivation
The world of algebraic groups, and even more generally affine group schemes, is a vast and scary place; all kinds of unexpected things can happen. Even in the cases that we best understand, like reductive groups over local fields, we are far from being able to ‘list’ all such possible groups (the best we can do is try and list root data in the reductive case).
That said, the vista isn’t so bleak that in the absolutely simplest of all cases, dimension (connected) algebraic groups over an algebraically closed field, we can’t say anything. In fact, quite the opposite. There are, up to isomorphism, only two such groups: the multiplicative group
, and the additive group
.
This result is very useful, not only in the sense of comfort it gives in the chaotic world of groups (“well, at least I know that case!”), but as a means of limiting the open affine subgroup schemes of some given object. Moreover, in certain nice situations, it will even allow us to classify one-dimensional connected algebraic groups over non-closed fields.
The setup
Before we start the proof, let’s take a moment to set conventions. Let us fix an algebraically closed field (of arbitrary characteristic). An algebraic group
is smooth affine group scheme over
(i.e. a smooth, closed subgroup scheme of
for some
).
Our goal is to show that if is a connected algebraic group of dimension
, then
must be one of the two simplest possible group schemes: the multiplicative group
, or the additive group
. Recall their definitions:
The multiplicative group: The multiplicative group has the obvious underlying scheme structure of the punctured line:
. The group structure on
has multiplication map
given by
inverse map given by
and identity section
Functorially, represents the group functor
given by
.
The additive group: The additive group has the underlying scheme structure of the line
. The addition map
is given by
inversion map given by
and identity section given by
Functorially, is the group functor sending a
-scheme
to
.
While I am sure that the vast majority of readers were well aware of those definitions, it’s nice to keep them in mind for later, when we do a fine structural analysis of the group structures on the line and punctured line.
The proof
Before we jump right into the formalities of the proof, let’s take a quick second to outline its major steps:
- Open embed our group
into an integral smooth projective curve
.
- Show, that using the group structure of
that
contains an infinite subset, and, moreover, this subset preserves the
-points of
.
- Use Hurwitz’s automorphism theorem, along with basic facts about elliptic curves, to deduce that the genus of
can’t be greater than
, and can’t be
respectively. Thus, deduce that
.
- Note that since
, has an infinite subset which stabilizes the set of
-points in
, that
can contain only
or
-points.
- Deduce that, as a scheme,
is isomorphic to the line or the punctured line.
- Show that all group structures on the line and punctured line give algebraic groups isomorphic to either
or
.
Now that we have this rough outline, let’s fill in the details:
Theorem(Main) 1: Let
be a connected algebraic group of dimension
. Then,
is isomorphic to either
or
.
Proof: Begin by noting that is automatically integral. Indeed, since
is Noetherian, connected, and has local rings which are domains (since it’s regular), the standard trick shows that if it were reducible, and the components intersected, one could push this intersection to the local ring at the intersection point, contradicting the local ring being integral. Now, since
is integral and regular, we know (Theorem 17.4.2 of Vakil) that we can open embed
into a smooth integral projective curve
.
For each we have the translation operator
. By the curve-to-extension theorem (cf. Vakil), each translation automorphism extends to a morphism
. But, since this map is obviously of degree
, it must be an automorphism of
. Thus, we obtain an injection
. Moreover, since
, and
, we know that
is infinite, and thus
is also infinite.
Now, we claim that this implies that the genus of is
, so that
. It is a famous theorem of Hurwitz, that if
is a curve over an algebraically closed field, and the genus of
is at least
, then
is finite. Thus, we certainly know that
has genus at most
. So, we only need to discount the case that the genus of
is
.
To do this, let us suppose that is of genus
, and choose a
-point
.. Note that we have a map
which respects composition. In particular, note that, since
is finite,
is the identity map infinitely often. In particular, there are infinitely many automorphisms of
which fix
.
The pair is an elliptic curve over
, and so there exists a unique
-group scheme structure on
such that
is the identity point. We then obtain infinitely many automorphisms of
which fix
, and thus an infinite number of elements of
(here
denotes self-isogenies). But, it’s a common fact that this set is finite (in fact, it’s of order at most
–cf. Theorem 10.1 of Silverman). Thus, we arrive at a contradiction.
So, at this point we know that , and we have an open embedding
. Moreover, we have an infinite set of automorphisms of
stabilizing the finite set
. But, if
consisted of more than
-points, then the permutation of
would determine the automorphism of
(this is just the classical theory of linear fractional transformations), meaning there could only be finitely many automorphisms coming from
. Thus,
consists of at most two
-points. Since
and
for any
-points
we may conclude that, as schemes,
is isomorphic to either
or
.
Thus, it remains to show what the possible algebraic group structures on and
are.
We begin by showing that, up to isomorphism of algebraic groups, the only algebraic group with underlying scheme structure is
. To see this, let
be an algebraic group with underlying scheme structure
. By changing
up to an isomorphism, we may assume without loss of generality that
has identity at
. Now, it suffices to show that for each
, the translation map
is just the map
. To see this, note that
is an automorphism of
(as a scheme) with no fixed points. A quick check shows that it’s of the form
for some
. But, since
, we may conclude that
.
Similarly, let be a group scheme with underlyling scheme structure of
. By the same argument as before, we may assume without loss of generality that the identity element of
is at
. The automorphisms of
are of the form
, for
. Now, let
, as before, we want to show that
is the automorphism
. Now, since
has no fixed points it must be of the form
(the map
has the fixed point
). But, then since it must preserve
, we see that
, and so the conclusion follows.
A simple application
Commutativity of one-dimensional algebraic groups
Before we discuss actual applications, let’s discuss some ‘weaker’ corollaries, which are in and of themselves surprising. If some person came up to you on the street and said “Hey, buddy, is every connected algebraic group of dimension over an algebraically closed field commutative?”, I don’t know what you’d say. It’s not at all obvious this is true, although we could conclude, a posteriori, from the above that this is true. Of course, you could try and prove this Lie theoretically (using the Lie algebra), but Lie theory for algebraic groups gets weird over positive characteristic.
Moreover, we actually get the commutativity of connected algebraic groups of dimension , even over non-closed fields:
Theorem 2: Let
be any field (not necessarily algebraically closed!), the any connected one-dimensional algebraic group over
is abelian.
Proof: Note we can check commutativity on -points. Indeed, let
be our connected, one-dimensional algebraic group. We want to show that the following identity of morphisms holds:
where is the obvious switching of coordinates map. Well, since
and
are extremely nice (in particular, reduced and separated), we can check this equality over
, where the result follows from the main theorem.
Restriction on open group subschemes of curves
The above theorem puts a serious restriction on what schemes can contain algebraic groups as (proper) open subschemes. The particular example I have in mind is the following.
Theorem 3: Suppose that
is a smooth, projective, geometrically integral curve (here I am NOT assuming that
!), and that
is not
. Then, no (proper) open subscheme of
can have the structure of a
-group scheme.
Proof: Indeed, suppose that is such an open subscheme. Then,
is an open subscheme of
, with the structure of a
-group scheme. Note though that
is necessarily affine. Indeed:
Lemma 4: Let
be a smooth, projective, geometrically integral curve. Then, any proper open subscheme
is affine.
Proof: If is empty, we’re done. So, assume that
is not empty. Then,
for some
-points
. Consider then the line bundle
, where
for
.
Now, choosing the sufficiently large, we know that
will have negative degree, and so we may assume that
. So, by choosing the
sufficiently large, we can made
sufficiently large. In particular, we can find
such that
has poles precisely at the
. Thus, we obtain a non-constant rational map
off of the
. Extending this to a morphism
, we obtain a finite map with
, and thus
is affine.
So, by our classification, we have that is either
or
. So, if
is birational to
in either case, and so (since
was assumed smooth), we have that
. Thus,
is a Brauer-Severi variety, and since we assumed that
had a
-point (since
had a
-point, being a group scheme!) we may conclude that
.
Now, this theorem is certainly overkill in most situations, but it does quickly allow one to process/internally verify various statements one often times hears. For example, take the statement
Let
, and let
be the closure of
inside of
. Then,
has a group structure, but
does not.
(this was, if you are curious, paraphrased from Darmon’s Rational Points on Curves). Now, while there may be other ways of verifying this, it also follows immediately from our above analysis.
What goes wrong over non-closed fields
There are various places in the argument for the main theorem that, in an important way, used that our base field was algebraically closed. Just in case you are still not convinced that we couldn’t have tweaked the proof to work in general, we do the most explicit thing we can to convince you. Namely, we give an example of a connected algebraic group which is neither
nor
.
As we shall see in the next section, we’re better off looking for things which geometrically are .
To begin, let’s recall the general formalism of Weil restriction. Namely, let be a finite separable extension of fields. Then, for any group scheme
we can form a group functor
, called the Weil restriction of
, as follows:
for every -algebra
. If the Weil restriction
is representable, we also call the representing scheme the Weil restriction of
, and also denote it
.
It is a general theorem that for finite separable, and an algebraic group
, the Weil restriction
, is representable by an algebraic group over
. For more information, see Raynaud et. al’s Neron Models. For example, a vast generalization of the above representability statement can be found as Theorem 4 in section 7.6.
Now, the notion of Weil restriction allows us to produce many examples of one-dimensional algebraic groups (obviously over non-closed fields) which are not or
.
Consider for example the connected algebraic group (I’m now decorating the multiplicative group with what field I’m considering it over). I can then consider the Weil restriction
. Note that this is NOT a one-dimensional algebraic group. In fact, the following is fairly easy to show:
Theorem 5: Let
be a finite Galois extension, and
an algebraic group. Then:
where
(here the notation means that in the fiber product the map
is the usual one and the map
is that induced by
). In particular, if
for some group
we have a canonical isomorphism of
-group schemes
and thus the above reduces to
In particular, we see that in our specific case, is two-dimensional.
But, we can cut down on the dimension by finding various subgroup schemes of . In particular, note that we have a canonical map
. Indeed, for each
-algebra
we have the map
which is just the usual norm map of algebras, considering the natural algebra map (note that the norm map makes sense since
is a free
-module!).
Let us denote the connected algebraic group by
. Note that we have the following short exact sequence of algebraic groups
Moreover, it’s easy to check that is indeed a connected algebraic group, since (referencing the theorem stated earlier)
then one can check directly (by considering the above exact sequence base changed to )
Thus, all we have left to check is that . But, this is clear since there are certainly
-algebras
such that
is not isomorphic to the ‘norm one’ elements of
.
For a truly simple example, take . Then, we have the following short exact sequence of groups
In particular
whereas .
Twists, and strengthenings of the theorem
It would be nice if we could somehow extend the above theorem to non-algebraically closed fields . Of course, as the example in the last section shows, this is not always possible in the most naive way. That said, there are several further observations which allow us to tighten the results of theorem, sometimes giving us a very practical, useful list of the possible one-dimensional algebraic groups over
.
Just to lay down some terminology, note that if is any field (not necessarily algebraically closed!) and
is a one-dimensional connected algebraic group over
, then
is either
or
. To see this we merely apply our main theorem and:
Observation: Let
be a finite type
-scheme and suppose that
. Then,
is connected if and only if
is connected. In particular, for a (finite type) group scheme over
connected implies geometrically connected.
Proof: Evidently if is connected then so is
(since we have a continuous surjection
). Suppose now that
is connected. Note that for an extension
one has that
is disconnected if and only if
contains non-trivial idempotents. In particular, since every element of
lies in
(note that we’re applying `flat base change’ here) for some finite extension
we see that
disconnected implies that
is disconnected for some finite extension
.
That said, note that is finite and flat (since
is finite and flat) and thus a clopen map. In particular, since
is connected we see that every connected component of
surjects on to
. In particular, fixing
we see that every connected component of
contains a preimage of
. But, since
is a
-point it has only one preimage. Since connected components are disjoint this implies that
has only one connected component. The conclusion follows.
Recall that a twist of a group scheme over a field
is a group scheme
for which
. So the last paragraph says, in other words,
is a twist of either the additive or multiplicative group. We shall call a one-dimensional algebraic group
of additive type if
, and of multiplicative type if
. Another common terminology for a group of multiplicative type is a one-dimensional torus, but we shall not use that terminology again.
We begin with the theorem which greatly reduces the amount of one-dimensional algebraic groups:
Theorem 6: Let
be a perfect field. Then, the only one-dimensional connected algebraic group
of additive type is
.
Proof: Embed into a regular curve
. Since
is perfect, and geometric regularity (i.e. smoothness) can be checked on perfect closures, we know that
is actually smooth. But,
, and thus, we see that
(since they are birational smooth curves). But, note that
removed two points cannot be
, and so
is one point. Moreover, this point
must be a
-point, else the standard short exact sequence
would be contradicted. But, note that since is a Brauer-Severi variety (see earlier), with a
-point, it must be
. Moreover, since
must base-change to the
-point
,
must consist of a single
-point as well. Thus,
, as a scheme, is
. Thus, previous arguments show that it must, in fact, be isomorphic to
as desired
Remark: Note the important place that this fails for . In the argument of the main theorem, we proved a statement about
and
having only one group structure up to isomorphism. We implied, and this can be easily seen, that this carries over to work over non-closed fields, but only for
. For
we used the closedness of
to discount the automorphism
as being a possible translation operator.
Thus, over a perfect field, all we have are , and groups of multiplicative type (i.e. tori).
There is a nice way of classifying these though. Namely, for any connected algebraic group , it’s common knowledge that the (isomorphism classes over
of) twists of
are just “
-torsors” (here
denotes the category of linear algebraic groups), and so are classified by
where, here, denotes the absolute Galois group
, and
denotes continuous group cohomology. We are thinking of
acting on
by just acting on the factor of
.
While this seems very scary, it’s actually reasonably manageable in most specific instances. For example, let’s see what happens when we take to be
, so that this should be classifying groups of multiplicative type. Well,
, with the actual automorphisms being
. Moreover, since neither of these automorphisms has any constant coefficients,
acts trivially. Thus, we see that
where the second to last step was because was a trivial
-module and the last step is by Kummer theory.
Note, while this certainly seems a bit strange at first, it is not at all to be unexpected. Indeed, consider our remark following the last theorem. The impediment to proving that the groups of multiplicative type over were just
came down to the automorphisms
possibly being fixed point free, which might give us different group structures on
. But, the automorphisms of those form with no fixed points correspond precisely to
with
, or, in other words, extensions of
of degree
!
As an example of this theory, we can see that there are precisely two groups of multiplicative type over , corresponding to the trivial extension, and to the extension
. Or, if one doesn’t want to use Kummer theory, we can stop at the step before, and calculate
But, since is procyclic, generated by
, any continuous homomorphism is determined by where
goes. Thus, there are two such continuous homomorphisms
. Moreover, from the last section, we know precisely what these two connected algebraic groups must be: the multiplicative group
and
.
So, to sum up this last part, together with our theorem of algebraic groups of additive type over perfect fields, we have the following tidy theorem:
Theorem 7: The only connected one-dimensional algebraic groups over
are
,
, and
.
Reduction types of elliptic curves
Using the classification of the last section, we can give a more conceptual understanding of the various ‘reduction types’ of an elliptic curve where
is a
-adic local field. Let us say that
is the residue field of
, a finite extension of
.
Now, there are various ways of defining the reduction over
. We shall take the low-brow approach here. Namely, to any Weierstrass equation
for
we can consider a minimal Weierstrass equation
which is defined to be a Weierstrass equation with coefficients in
whose discriminant has minimal valuation. We shall denote the curve associated to
by
. Note that
is a curve contained in
.
One then defines the reduction of , denoted
, to be
, a curve over
. One then begins to partition the set of
into classes based on properties of
. The most crude sorting of this type is made by saying that
has good reduction if
is an elliptic curve (i.e. whether
) and that it has bad reduction otherwise.
But, a further reduction is made based off of the smooth locus of . Before we define this, let us recall the following simple lemma:
Lemma 8: For any field
a cubic curve
can have at most one singularity, and it’s defined over
.
Proof: Suppose first that , and that
had two singularities
. Take a line
passing through both
and
. Consider then the intersection
(thought of as the intersection of divisors in the sense of intersection theory). Note that each of
and
cannot have intersection multiplicity
, else
and
would be smooth points of
(since their maximal ideal at those points would be singly generated).
Thus, we may conclude that
But, by Bezout’s theorem we have that which is a contradiction.
Assume now that . If
had two singular points they would stay singular over
contradicting the previous case. Similarly, even if
had one singular point, but it was not defined over
, then
would pick up at least two singular points over
.
So, from this we see that if has bad reduction, then it has just one singular point. Working a bit harder, one can show that either
is essentially ‘cuspidal’ (i.e. isomorphic to
) or ‘nodal’ (i.e. isomorphic to
). In the former case we say that
has additive reduction, and in the former case we say that
has multiplicative reduction.
We make the classification even more granular by the following sort of vague statement. Namely, imagine a nodal curve. Then, approaching the singular point there are two ‘branches’, and thus the singular point has two distinct tangent lines at the nodal point. We say that has split multplicative reduction if these tangent lines are defined over
, and non-split multiplicative reduction otherwise.
The more rigorous statement is that geometrically is a nodal curve. Then,
has split multiplicative reduction if it is actually a nodal curve over
, and non-split if its only nodal over some finite extension (one can show that it actually always happens over the quadratic extension of
).
To me, while these definitions literally make sense, they are a bit cumbersome, and counterintuitive. How did one think of these definitions? What is really happening here? Using the last section we can make better definitions of these terms and, in fact, justify the definitions above (e.g. why geometrically is either cusidpidal or nodal).
Namely, let us consider the reduction , and define the smooth locus of
, denoted
, to be the set of smooth points of
. By Lemma 8 we know that
is either
or
for some
. Now, using the exact same chord-tangent construction in the classical theory of elliptic curves, one can show that
possesses the structure of an algebraic group.
In particular, by Lemma 4, we know that is a connected affine algebraic group over
, and thus by Theorem 7 we know that
is either
,
, or
(where
is the quadratic extension). Thus, we have the following definition/theorem:
Theorem/Definition 9: Let
be an elliptic curve. Then,
has good (resp. additive, resp. split multiplicative, resp. non-split multiplicative) reduction if
is an elliptic curve (resp.
, resp.
, resp.
).
This is, in my opinion, the best version of the definitions of these terms. But, one can do a pretty simple case analysis to see that this definition is equivalent to the previous ones.
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