From a5d66cb6da3f6ab3b79898769b4d118c8b9e498b Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Sat, 6 May 2023 17:43:11 +0200 Subject: Some changes and additions --- src/research/script-kummer-kwg.txt | 224 ------------------------------------- 1 file changed, 224 deletions(-) delete mode 100644 src/research/script-kummer-kwg.txt (limited to 'src/research/script-kummer-kwg.txt') diff --git a/src/research/script-kummer-kwg.txt b/src/research/script-kummer-kwg.txt deleted file mode 100644 index 8759ee7..0000000 --- a/src/research/script-kummer-kwg.txt +++ /dev/null @@ -1,224 +0,0 @@ ---- 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. - - -- cgit v1.3