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a/src/research/notes-kummerec.pdf and /dev/null differ diff --git a/src/research/research.md b/src/research/research.md index 20cd1fe..ca5b4a0 100644 --- a/src/research/research.md +++ b/src/research/research.md @@ -5,10 +5,11 @@ After obtaining a bachelor degree in Trento in 2016 I have completed the special focus on number theory and algebraic geometry. I then moved on to obtain a joint PhD at the universities of Luxembourg and Leiden, under the supervision of [Antonella Perucca](http://antonellaperucca.net) and -[Peter Bruin](https://www.math.leidenuniv.nl/~pbruin). I have collected -in this page the list of papers I puplished during my PhD, as well as -slides or notes for all the talks I have given and my theses (Bachelor, -Master, PhD). +[Peter Bruin](https://www.math.leidenuniv.nl/~pbruin). + +I have collected in this page the list of papers I puplished during my PhD, +as well as theses (Bachelor, Master, PhD). See [my talks page](../talks) +for a list of slides and notes of my talks. ## Theses @@ -69,100 +70,3 @@ Master, PhD). * Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano. *Explicit Kummer theory for the rational numbers.* International Journal of Number Theory (2020). - -## Talks - -* *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: [link](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)] 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. - - diff --git a/src/research/slides-alggroupsfieldext.pdf b/src/research/slides-alggroupsfieldext.pdf deleted file mode 100644 index 2b48adb..0000000 Binary files a/src/research/slides-alggroupsfieldext.pdf and /dev/null differ diff --git a/src/research/slides-ecm.pdf b/src/research/slides-ecm.pdf deleted file mode 100644 index 9305984..0000000 Binary files a/src/research/slides-ecm.pdf and /dev/null differ diff --git a/src/research/slides-fieldextec.pdf b/src/research/slides-fieldextec.pdf deleted file mode 100644 index 6b54ff7..0000000 Binary files a/src/research/slides-fieldextec.pdf and /dev/null differ diff --git a/src/research/slides-groupcohomec.pdf b/src/research/slides-groupcohomec.pdf deleted file mode 100644 index 72a2a3c..0000000 Binary files a/src/research/slides-groupcohomec.pdf and /dev/null differ diff --git a/src/research/slides-journey-kummer.pdf b/src/research/slides-journey-kummer.pdf deleted file mode 100644 index 9edf804..0000000 Binary files a/src/research/slides-journey-kummer.pdf and /dev/null differ diff --git a/src/research/slides-kummec-barcelona.pdf b/src/research/slides-kummec-barcelona.pdf deleted file mode 100644 index 7af439d..0000000 Binary files a/src/research/slides-kummec-barcelona.pdf and /dev/null differ diff --git a/src/research/slides-kummec-diamant.pdf b/src/research/slides-kummec-diamant.pdf deleted file mode 100644 index d66f002..0000000 Binary files a/src/research/slides-kummec-diamant.pdf and /dev/null differ diff --git a/src/research/slides-kummer-diamant.pdf b/src/research/slides-kummer-diamant.pdf deleted file mode 100644 index 2160eac..0000000 Binary files a/src/research/slides-kummer-diamant.pdf and /dev/null differ diff --git a/src/research/slides-kummer-final.pdf b/src/research/slides-kummer-final.pdf deleted file mode 100644 index e24e01a..0000000 Binary files a/src/research/slides-kummer-final.pdf and /dev/null differ diff --git a/src/research/slides-kummer-kwg.pdf b/src/research/slides-kummer-kwg.pdf deleted file mode 100644 index 3940f4a..0000000 Binary files a/src/research/slides-kummer-kwg.pdf and /dev/null differ diff --git a/src/research/slides-kummerdegrees.pdf b/src/research/slides-kummerdegrees.pdf deleted file mode 100644 index 2b13012..0000000 Binary files a/src/research/slides-kummerdegrees.pdf and /dev/null differ diff --git a/src/research/slides-local-global.pdf b/src/research/slides-local-global.pdf deleted file mode 100644 index 32a84fd..0000000 Binary files a/src/research/slides-local-global.pdf and /dev/null differ diff --git a/src/research/slides-tronto-bristol.pdf b/src/research/slides-tronto-bristol.pdf deleted file mode 100644 index 4ece0b7..0000000 Binary files a/src/research/slides-tronto-bristol.pdf and /dev/null differ -- cgit v1.3