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