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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. + + diff --git a/src/talks/slides-alggroupsfieldext.pdf b/src/talks/slides-alggroupsfieldext.pdf new file mode 100644 index 0000000..2b48adb Binary files /dev/null and b/src/talks/slides-alggroupsfieldext.pdf differ diff --git a/src/talks/slides-ecm.pdf b/src/talks/slides-ecm.pdf new file mode 100644 index 0000000..9305984 Binary files /dev/null and b/src/talks/slides-ecm.pdf differ diff --git a/src/talks/slides-fieldextec.pdf b/src/talks/slides-fieldextec.pdf new file mode 100644 index 0000000..6b54ff7 Binary files /dev/null and b/src/talks/slides-fieldextec.pdf differ diff --git a/src/talks/slides-groupcohomec.pdf b/src/talks/slides-groupcohomec.pdf new file mode 100644 index 0000000..72a2a3c Binary files /dev/null and b/src/talks/slides-groupcohomec.pdf differ diff --git a/src/talks/slides-journey-kummer.pdf b/src/talks/slides-journey-kummer.pdf new file mode 100644 index 0000000..9edf804 Binary files /dev/null and b/src/talks/slides-journey-kummer.pdf differ diff --git a/src/talks/slides-kummec-barcelona.pdf b/src/talks/slides-kummec-barcelona.pdf new file mode 100644 index 0000000..7af439d Binary files /dev/null and b/src/talks/slides-kummec-barcelona.pdf differ diff --git a/src/talks/slides-kummec-diamant.pdf b/src/talks/slides-kummec-diamant.pdf new file mode 100644 index 0000000..d66f002 Binary files /dev/null and b/src/talks/slides-kummec-diamant.pdf differ diff --git a/src/talks/slides-kummer-diamant.pdf b/src/talks/slides-kummer-diamant.pdf new file mode 100644 index 0000000..2160eac Binary files /dev/null and b/src/talks/slides-kummer-diamant.pdf differ diff --git a/src/talks/slides-kummer-final.pdf b/src/talks/slides-kummer-final.pdf new file mode 100644 index 0000000..e24e01a Binary files /dev/null and b/src/talks/slides-kummer-final.pdf differ diff --git a/src/talks/slides-kummer-kwg.pdf b/src/talks/slides-kummer-kwg.pdf new file mode 100644 index 0000000..3940f4a Binary files /dev/null and b/src/talks/slides-kummer-kwg.pdf differ diff --git a/src/talks/slides-kummerdegrees.pdf b/src/talks/slides-kummerdegrees.pdf new file mode 100644 index 0000000..2b13012 Binary files /dev/null and b/src/talks/slides-kummerdegrees.pdf differ diff --git a/src/talks/slides-local-global.pdf b/src/talks/slides-local-global.pdf new file mode 100644 index 0000000..32a84fd Binary files /dev/null and b/src/talks/slides-local-global.pdf differ diff --git a/src/talks/slides-tronto-bristol.pdf b/src/talks/slides-tronto-bristol.pdf new file mode 100644 index 0000000..4ece0b7 Binary files /dev/null and b/src/talks/slides-tronto-bristol.pdf differ diff --git a/src/talks/talks.md b/src/talks/talks.md new file mode 100644 index 0000000..b7cea5b --- /dev/null +++ b/src/talks/talks.md @@ -0,0 +1,119 @@ +# Talks + +## Cubing + +* *Cubo di Rubik e Speedcubing (ITA)*. + Invited talk at Università di Cagliari, January 2020. + [Slides: [pdf, 2.0Mb](cubo-cagliari.pdf)] + +* *The Rubik's cube group structure in 2 minutes*. + Speed talk at the AAN Symposium, May 2019. + [Slides: [pdf, 1.2Mb](cube-2min-slides.pdf)] + +## Math + +* *Kummer theory for elliptic curves*. + Short talk for Tarrach Prize 2023, March 2023. + [Slides: [pdf, 288Kb](kummer-tarrach.pdf)] + +* *Kummer theory for commutative algebraic groups*. + Seminar at the university of Leiden, September 2022. + [Slides: [pdf, 1.2Mb](slides-kummer-final.pdf)] + +* *My journey in Kummer theory*. + Leiden-Luxembourg Number Theory PhD Days 2022, July 5-6. + [Slides: [pdf, 521Kb](slides-journey-kummer.pdf)] + +* *A tour of group theory with our companion cube*. + University of Luxembourg PhD seminar, May 2022. + [Slides: [pdf, 1.1Mb](group-cube.pdf)] + +* *Kummer theory via (J,T)-extensions*. + University of Leiden algebra seminar, May 2022; + similar to *A category of division modules*, see below. + +* *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 + *Division in modules and Kummer theory*, see below. + [Notes: [pdf, 245Kb](division-leiden.pdf)] + +* *Division in modules and Kummer theory*. + Invited talk at the University of Groningen, February 2022. + [Slides: [pdf, 249Kb](division-groningen.pdf)] + +* *A category of division modules*. + University of Luxembourg Number Theory seminar, November 2021. + [Notes: [pdf, 270Kb](notes-division-modules.pdf)] + +* *Introduction to étale cohomology*. + University of Luxembourg seminar series on perverse sheaves, October 2021. + [Notes: [pdf, 237Kb](notes-etale.pdf)] + +* *A generalization of injective modules*. + University of Luxembourg Number Theory seminar, October 2021. + [Notes: [pdf, 259Kb](notes-injectivity.pdf)] + +* *Integer factorization and elliptic curves*. + University of Leiden Bachelor seminar (guest talk), April 2021. + [Slides: [pdf, 393Kb](slides-ecm.pdf)] + +* *Group cohomology and elliptic curves*. + Invited talk at Number Theory Online conference, February 2021. + [Slides: [pdf, 210Kb](slides-groupcohomec.pdf); + [Video](https://vimeo.com/526814236/1a640285c0?embedded=true&source=video_title&owner=47245911)] + +* *Kummer theory for algebraic groups*. + Invited talk at the University of Bristol (online), October 2020. + [Slides: [pdf, 239Kb](slides-tronto-bristol.pdf)] + +* *Algebraic groups and field extensions*. + University of Luxembourg PhD seminar (online), April 2020. + [Slides: [pdf, 280Kb](slides-alggroupsfieldext.pdf)] + +* *Field extensions and elliptic curves*. + University of Luxembourg PhD Day, October 2019. + [Slides: [pdf, 6.0Mb](slides-fieldextec.pdf)] + +* *Kummer theory for elliptic curves*. + University of Leiden algebra seminar, September 2019; + repeated with slides at the DIAMANT symposium, November 2019 and + at Seminari de Teoria de Nombres de Barcelona, February 2020. + [Notes: [pdf, 181Kb](notes-kummerec.pdf); + Slides DIAMANT: [pdf, 271Kb](slides-kummec-diamant.pdf); + Slides Barcelona: [pdf, 290Kb](slides-kummec-barcelona.pdf)]. + +* *Divisibility of points in algebraic groups*. + Invited talk at the University of the Basque Country, May 2019. + [Notes: [pdf, 231Kb](notes-bilbao.pdf)] + +* *Kummer theory for number fields*. + Fifth symposium of the Roman Number Theory association, April 2019; + repeated at the Algant Alumni symposium in Benasque, May 2019. + [Slides: [pdf, 291Kb](slides-kummerdegrees.pdf)] + +* *The local-global principle*. + University of Luxembourg interdisciplinary seminar, March 2019. + [Slides: [pdf, 598Kb](slides-local-global.pdf)] + +* *Local-global principle for torsion*. + University of Luxembourg Number Theory seminar, November 2018. + [Notes: [pdf, 208Kb](notes-katz.pdf)] + +## Software + +* *L'importanza del software libero nella ricerca (ITA)*. + Linux day in Feltre, October 2019. + [Slides: [pdf, 804Kb](linuxday2019.pdf); + [Video](https://www.youtube.com/watch?v=FB0YIYNp3rw)] + +* *Linux from scratch (ITA)* + Linux day in Sedico, October 2012. + [Video](https://www.youtube.com/watch?v=h7xW7iq-ApA). -- cgit v1.3