commit 1dbba9485536a174358ded5f9c84d828994c7845
parent d6057b95b0a91e70957a1f3618380d3230403f97
Author: Sebastiano Tronto <sebastiano@tronto.net>
Date: Wed, 20 Apr 2022 15:49:40 +0200
Added two talks
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diff --git a/src/research/research.md b/src/research/research.md
@@ -50,6 +50,13 @@ under the supervision of [Antonella Perucca](http://antonellaperucca.net) and
## Talks
+* *Kummer theory for algebraic groups*
+ Nederlands Matematisch Congres, April 2022;
+ repeated at the DIAMANT symposium, April 2022.
+ [Notes (NMC): [pdf, 215Kb](slides-kummer-kwg.pdf);
+ Script (NMC): [txt, 9Kb](script-kummer-kwg.txt);
+ Notes (DIAMANT, longer version): [pdf, 223Kb](slides-kummer-diamant.pdf)]
+
* *Division in modules*.
University of Leiden algebra seminar, April 2022;
similar to *A generalization of injective modules* and
diff --git a/src/research/script-kummer-kwg.txt b/src/research/script-kummer-kwg.txt
@@ -0,0 +1,224 @@
+--- 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.
+
+
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