--- Kummer theory ---
0:50
Kummer theory is the study of field extensions generated by the n-th roots
of elements of a base field, such as the rational numbers. Taking all
n-th roots ensures that such an extension is Galois and that it contains
the cyclotomic field: indeed, one can write the n-th roots of unity as
ratios of different n-th roots of the same element.
To study these extensions it is convenient to start not just with a
set of elements, but with a multiplicative subgroup of the base field.
For example, one can take the group generated by one element. This does
not change anything on the field-theoretic side: the extensions we are
considering do not change. But it makes the "purely algebraic" side of
things more convenient: the group sqrt[n]{A} now is group which contains
A and the roots of unity.
--- Kummer theory for algebraic groups ---
1:00
"Kummer theory for algebraic groups" is a similar, more general
problem. If we take G to be a commutative algebraic group over a number
field K, we can take a subgroup of the K-rational points of G and you
consider the n-division points of this group, which is an analogue of
the group of n-th roots in the previous case.
If you add the coordinates of these points to your base field you obtain
what a field extension which has properties remarkably similar to the
classical Kummer extensions: it is Galois over K and it contains the
n-torsion field of G, an analogue of the cyclotomic field generated by
the torsion points of G.
This is a generalization of the classical case, because if you take G
to be the multiplicative group you obtain exactly that case.
These field extensions are the kind of objects that I am studying.
--- Results for elliptic curves ---
1:50
So, what do we want to know about these kind of field extensions? One of
the things we care about is estimating, or computing, their degrees.
This is because they have applications in other areas of number theory:
when studying problems related to Artin's primitive root conjecture it
can happen that the density of certain set of primes can be expressed
in terms of the degrees of Kummer extensions.
The degree of a Kummer extension over the torsion field is always
between a certain power of n and the same power of n times a constant.
This power of n is for example 2 in the case of elliptic curves and
groups of points generated by one non-torsion element. So they cannot
be much smaller than the maximum.
In recent years there has been effort in making these results effective.
In a recent work with Lombardo we were able to quantify this constant, or
a possible value for it, in terms of computable properties of the curve
and of the chosen point, for curves without complex multiplication. In
particular, one of these properties are the p-adic Galois representation
associated with the curve. Over Q we even have an explicit and uniform
estimate for such a constant.
In our work we had problems when the curve had non-trivial endomorphisms
defined over the base field, so CM curves. However Abtien Javan Peykar, a
student of Lenstra, managed to get similar results for CM curves only. So,
how did he manage?
--- Endomorphism rings ---
0:55
The problem Lombardo and I had was caused by considering the A and its
division groups as abelian groups. Instead, Javan Peykar decided to
take them as modules over the endomorphism ring of the curve, an order
in a quadratic imaginary field - and even with some extra technical
limitations, such as considering only maximal orders.
So my idea was: if we can do the same over a *general* ring, regardless
of it being Z or an order in a number field or anything else, maybe we
can build a general framework to study these division groups, or rather
division modules, and then apply all of this to do Kummer theory over
other classes of algebraic groups.
So this is what I did.
--- Division modules ---
1:00
The first part is understanding what "division in modules is", starting
from the "denominator": what do we divide by?
As is often the case in commutative algebra, it is convenient to use,
instead of the elements of the ring, ideals of the ring. So for M
contained in N we define the I-division module of M inside N to be the
set of elements of N that multiplied by I end up inside M. This is a
classical definition that is found in some commutative algebra books.
We also want to consider infinite unions of such division modules. For
example we might want to work with the set of all division points of
our subgroup of rational points. In our first example, all n-th roots
of a certain number.
To do this, we introduce the concept of "ideal filter", which like a
fiter in set theory but for ideals.
--- Ideal filters ---
0:20
In practice I always want to divide by one of these two families of
ideals: either the one generated by all positive integers or the one
generated by powers of a given prime.
These are the main example of ideal filters that we should keep in mind,
but I will develop my theory in general.
--- J-injectivity ---
1:25
The set of all division points is a divisible abelian group. But over
a general ring this divisibility property can be awkward to work with,
and we prefer to use injectivity, which is equivalent to divisibility
over the ring Z.
A module is called injective when maps to it can be lifted along
injective morphisms. We call it instead J-injective when maps to it can
be lifted along certain injective morphisms, namely those such that the
codomain coincides with the module of J-division points of the image.
This definition captures the concept of "dividing only by J".
This a nice and simple generalization of a classical concepts, but it has
some noteworthy properties. First of all it is a true generalization:
taking J to be the set of all right ideals of R it becomes equivalent
to injectivity. And for example one can use it to extend the definition
of p-divisible abelian group: over Z p-divisibility is equivalent to
p^\infty-injectivity, where p^\infty is the ideal filter I introduced
in the previous slide.
One might say that this definition highlights the connection between
injectivity and divisibility better than the classical one does.
--- (J,T)-extensions ---
1:20
The last ingredient to complete our algebraic theory is the torsion. We
are building all this theory of division modules abstractly, in a way
independent of the agebraic group G that we started with. But when we do
this and we consider division modules, there is no way for this objects
to know that they are supposed to live in some elliptic curve rather than
in the multiplicative group or in some higher-dimensional abelian variety.
We need some extra structure. We need to fix a torsion and J-injective
module T that plays the role of the torsion subgroup of G.
Then we consider only those extensions of a base module M that consist
of division points and whose torsion embeds into T.
These objects form a category with many nice properties, that strongly
resembles the category of field extensions of a fixed field.
We also have an analogue of an algebraic closure, that plays the role
of the "set of all division points" that we have mentioned. This can be
constructed as a "J-hull", which is the analogue of the injective hull,
or injective envelope, for our generalization of injectivity.
With this category we can establish many properties of division modules,
and study their automorphisms.
--- Galois representations ---
0:55
And finally, how do I use this whole theory to study my number theoretical
problems? I can consider the Galois group of my Kummer extension,
say the one generated by all division points, and it embeds into the
automorphism group of this maximal (J,T)-extension.
This automorphism group fits into a short exact sequence.
Then the standard short exact sequence of Galois theory embeds into this
and we obtain this commutative diagram of groups, with exact rows. This
sequence is the main tool to study Kummer theory for algebraic groups.
Finding an explicit lower bound for the degrees I talked about amounts
to proving an explicit open image theorem for this "representation"
on the left-hand side.
In short, this diagram is the key to derive number-theoretic results
from certain key properties of the group.
--- New results ---
0:50
So, what kind of new results were we able to obtain with this technical
tools in our hands?
First of all, it was easy to unify the CM and non-CM cases and
show that one does not need to separate the two cases, except for
studying some specific properties of the curves related to their Galois
representations. Once you have have them, you can plug in any elliptic
curve with any endomorphism ring into our general framework and you
obtain the (already known) results. This also completes the CM case,
that had some missing pieces due to technical difficulties.
More importantly, in my opinion, we have now a better understanding of
these objects.
You see, when studying a problem cases by case is like you are trying
to find your way in a forest step by step. With this general framework
we have a way-better overview of the landscape we are moving in.
Lastly, the generality of this theory allows one to obtain some results
for higher-dimensional abelian varieties. This is work in progress,
but we already have results for some classes of varieties. There some
technical things to work out related to understanding the torsion subgroup
as a module over the endomorphism ring, but I am optimistic that we will
work this out.