 # sebastiano.tronto.net

Source files and build scripts for my personal website
git clone https://git.tronto.net/sebastiano.tronto.net

script-kummer-kwg.txt (9577B)

      1 --- Kummer theory ---
2 0:50
3
4 Kummer theory is the study of field extensions generated by the n-th roots
5 of elements of a base field, such as the rational numbers.  Taking all
6 n-th roots ensures that such an extension is Galois and that it contains
7 the cyclotomic field: indeed, one can write the n-th roots of unity as
8 ratios of different n-th roots of the same element.
9
10 To study these extensions it is convenient to start not just with a
11 set of elements, but with a multiplicative subgroup of the base field.
12 For example, one can take the group generated by one element.  This does
13 not change anything on the field-theoretic side: the extensions we are
14 considering do not change. But it makes the "purely algebraic" side of
15 things more convenient: the group sqrt[n]{A} now is group which contains
16 A and the roots of unity.
17
18
19 --- Kummer theory for algebraic groups ---
20 1:00
21
22 "Kummer theory for algebraic groups" is a similar, more general
23 problem. If we take G to be a commutative algebraic group over a number
24 field K, we can take a subgroup of the K-rational points of G and you
25 consider the n-division points of this group, which is an analogue of
26 the group of n-th roots in the previous case.
27
28 If you add the coordinates of these points to your base field you obtain
29 what a field extension which has properties remarkably similar to the
30 classical Kummer extensions: it is Galois over K and it contains the
31 n-torsion field of G, an analogue of the cyclotomic field generated by
32 the torsion points of G.
33
34 This is a generalization of the classical case, because if you take G
35 to be the multiplicative group you obtain exactly that case.
36
37 These field extensions are the kind of objects that I am studying.
38
39
40 --- Results for elliptic curves ---
41 1:50
42
43 So, what do we want to know about these kind of field extensions?  One of
44 the things we care about is estimating, or computing, their degrees.
45 This is because they have applications in other areas of number theory:
46 when studying problems related to Artin's primitive root conjecture it
47 can happen that the density of certain set of primes can be expressed
48 in terms of the degrees of Kummer extensions.
49
50 The degree of a Kummer extension over the torsion field is always
51 between a certain power of n and the same power of n times a constant.
52 This power of n is for example 2 in the case of elliptic curves and
53 groups of points generated by one non-torsion element. So they cannot
54 be much smaller than the maximum.
55
56 In recent years there has been effort in making these results effective.
57 In a recent work with Lombardo we were able to quantify this constant, or
58 a possible value for it, in terms of computable properties of the curve
59 and of the chosen point, for curves without complex multiplication.  In
60 particular, one of these properties are the p-adic Galois representation
61 associated with the curve. Over Q we even have an explicit and uniform
62 estimate for such a constant.
63
64 In our work we had problems when the curve had non-trivial endomorphisms
65 defined over the base field, so CM curves.  However Abtien Javan Peykar, a
66 student of Lenstra, managed to get similar results for CM curves only. So,
67 how did he manage?
68
69
70 --- Endomorphism rings ---
71 0:55
72
73 The problem Lombardo and I had was caused by considering the A and its
74 division groups as abelian groups. Instead, Javan Peykar decided to
75 take them as modules over the endomorphism ring of the curve, an order
76 in a quadratic imaginary field - and even with some extra technical
77 limitations, such as considering only maximal orders.
78
79 So my idea was: if we can do the same over a *general* ring, regardless
80 of it being Z or an order in a number field or anything else, maybe we
81 can build a general framework to study these division groups, or rather
82 division modules, and then apply all of this to do Kummer theory over
83 other classes of algebraic groups.
84
85 So this is what I did.
86
87
88 --- Division modules ---
89 1:00
90
91 The first part is understanding what "division in modules is", starting
92 from the "denominator": what do we divide by?
93
94 As is often the case in commutative algebra, it is convenient to use,
95 instead of the elements of the ring, ideals of the ring. So for M
96 contained in N we define the I-division module of M inside N to be the
97 set of elements of N that multiplied by I end up inside M. This is a
98 classical definition that is found in some commutative algebra books.
99
100 We also want to consider infinite unions of such division modules. For
101 example we might want to work with the set of all division points of
102 our subgroup of rational points. In our first example, all n-th roots
103 of a certain number.
104
105 To do this, we introduce the concept of "ideal filter", which like a
106 fiter in set theory but for ideals.
107
108
109 --- Ideal filters ---
110 0:20
111
112 In practice I always want to divide by one of these two families of
113 ideals: either the one generated by all positive integers or the one
114 generated by powers of a given prime.
115
116 These are the main example of ideal filters that we should keep in mind,
117 but I will develop my theory in general.
118
119
120 --- J-injectivity ---
121 1:25
122
123 The set of all division points is a divisible abelian group. But over
124 a general ring this divisibility property can be awkward to work with,
125 and we prefer to use injectivity, which is equivalent to divisibility
126 over the ring Z.
127
128 A module is called injective when maps to it can be lifted along
129 injective morphisms. We call it instead J-injective when maps to it can
130 be lifted along certain injective morphisms, namely those such that the
131 codomain coincides with the module of J-division points of the image.
132 This definition captures the concept of "dividing only by J".
133
134 This a nice and simple generalization of a classical concepts, but it has
135 some noteworthy properties. First of all it is a true generalization:
136 taking J to be the set of all right ideals of R it becomes equivalent
137 to injectivity. And for example one can use it to extend the definition
138 of p-divisible abelian group: over Z p-divisibility is equivalent to
139 p^\infty-injectivity, where p^\infty is the ideal filter I introduced
140 in the previous slide.
141
142 One might say that this definition highlights the connection between
143 injectivity and divisibility better than the classical one does.
144
145
146 --- (J,T)-extensions ---
147 1:20
148
149 The last ingredient to complete our algebraic theory is the torsion. We
150 are building all this theory of division modules abstractly, in a way
151 independent of the agebraic group G that we started with.  But when we do
152 this and we consider division modules, there is no way for this objects
153 to know that they are supposed to live in some elliptic curve rather than
154 in the multiplicative group or in some higher-dimensional abelian variety.
155
156 We need some extra structure. We need to fix a torsion and J-injective
157 module T that plays the role of the torsion subgroup of G.
158
159 Then we consider only those extensions of a base module M that consist
160 of division points and whose torsion embeds into T.
161
162 These objects form a category with many nice properties, that strongly
163 resembles the category of field extensions of a fixed field.
164
165 We also have an analogue of an algebraic closure, that plays the role
166 of the "set of all division points" that we have mentioned. This can be
167 constructed as a "J-hull", which is the analogue of the injective hull,
168 or injective envelope, for our generalization of injectivity.
169
170 With this category we can establish many properties of division modules,
171 and study their automorphisms.
172
173
174 --- Galois representations ---
175 0:55
176
177 And finally, how do I use this whole theory to study my number theoretical
178 problems?  I can consider the Galois group of my Kummer extension,
179 say the one generated by all division points, and it embeds into the
180 automorphism group of this maximal (J,T)-extension.
181
182 This automorphism group fits into a short exact sequence.
183
184 Then the standard short exact sequence of Galois theory embeds into this
185 and we obtain this commutative diagram of groups, with exact rows.  This
186 sequence is the main tool to study Kummer theory for algebraic groups.
187
188 Finding an explicit lower bound for the degrees I talked about amounts
189 to proving an explicit open image theorem for this "representation"
190 on the left-hand side.
191
192 In short, this diagram is the key to derive number-theoretic results
193 from certain key properties of the group.
194
195
196 --- New results ---
197 0:50
198
199 So, what kind of new results were we able to obtain with this technical
200 tools in our hands?
201
202 First of all, it was easy to unify the CM and non-CM cases and
203 show that one does not need to separate the two cases, except for
204 studying some specific properties of the curves related to their Galois
205 representations. Once you have have them, you can plug in any elliptic
206 curve with any endomorphism ring into our general framework and you
207 obtain the (already known) results. This also completes the CM case,
208 that had some missing pieces due to technical difficulties.
209
210 More importantly, in my opinion, we have now a better understanding of
211 these objects.
212
213 You see, when studying a problem cases by case is like you are trying
214 to find your way in a forest step by step. With this general framework
215 we have a way-better overview of the landscape we are moving in.
216
217 Lastly, the generality of this theory allows one to obtain some results
218 for higher-dimensional abelian varieties. This is work in progress,
219 but we already have results for some classes of varieties. There some
220 technical things to work out related to understanding the torsion subgroup
221 as a module over the endomorphism ring, but I am optimistic that we will
222 work this out.
223
224