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Iwasawa theory of Galois representations

Final Report Summary - IWTHEGAR (Iwasawa theory of Galois representations)

Project objectives

The objectives of the project IWTHEGAR, as stated in Annex 1 of the grant agreement, were two-fold:
A. The first set of objectives were to use the funds for IWTHEGAR for training the researcher, for the transfer of knowledge activities of the researcher (or, organised by the researcher) and to help the researcher integrate in the scientific community within Europe.
B. The second objective was to tackle and resolve the following (and related) mathematical problems of mainstream research.

(i) Main conjectures for big Galois representations attached to a family of Hilbert modular forms (Section 1.1.1 of Annex 1).
(ii) Extended Selmer groups, height pairings and Rubin's formula (Section 1.1.2 of Annex 1).
(iii) Kolyvagin systems of rank r (Section1.1.3 of Annex 1).
(iv) Kolyvagin systems of rank r (Section 1.1.4 of Annex 1).
(iv) Jochnowitz style congruences (Section 1.1.5 of Annex 1).

Dr Buyukboduk believes that all goals, as stated in the grant agreement, have been achieved. This shall be explained in what follows.

A. Reintegration activities
Dr Buyukboduk used his funds to participate in numerous workshops and conferences, which all played a critical role in his training as a researcher. These are detailed in Section 2 of the Final Report. This provided Dr Buyukboduk with an invaluable chance to discuss his results and his line of investigations with some of the leaders of the mathematical community. This way, he was able to establish himself as an active member of his research area. Within the framework of IWTHEGAR, Dr Buyukboduk has visited many leading research institutes based in Chile, Germany, Hong Kong, Japan, Morocco, UK, USA and Turkey. Overall, he has delivered 28 lectures in the duration of IWTHEGAR. He believes, thanks to these visits, he was able to share his most recent results with his colleagues in a very timely and efficient manner. Without a doubt, the funds provided by European Commission through IWTHEGAR have played in this regard a very significant and a very positive role in his reintegration process to the scientific community of Europe and to help him to keep his ties intact with the research groups based outside of Europe.

Dr Buyukboduk also used the funds of IWTHEGAR to host some leading researchers in his area of interest, such as David Burns (King's College London, UK), Jan Nekovar (Paris VI, France), Tadashi Ochiai (Osaka University, Japan), Robert Pollack (Boston University, USA), Karl Rubin (UC Irvine, USA), Shu Sasaki (King's College London, UK) and Fabien Trihan (University of Nottingham, UK). His visitors have delivered several lectures during their visits, which gave the local mathematical community a unique opportunity to interact with some of the leading researchers in the areas of arithmetic geometry and number theory, so as to get acquainted with mathematical problems and most recent results at its very frontier.

As of April 2012, Dr Buyukboduk took an initiative to organise monthly meetings (the 'Istanbul Number Theory Meetings', abbreviated as INTM) with the aim of bringing the number theorists based in Istanbul, Turkey to discuss their current research projects. This, he hopes, will enhance the interaction within the relatively small group of researchers in his local community. The first four meetings of INTM received enormous interest, not only of the senior members but also from the graduate students.

B. Work performed and results
Dr Buyukboduk has obtained the following results on the research problems proposed in Annex 1 of the grant agreement:
1. Main conjectures for big Galois representations attached to a family of modular forms (Section 1.1.1 of Annex 1): Dr Buyukboduk has finalised his proposed work so as to construct a Kolyvagin system out of Howard's Euler system of big Heegner points and thus prove a divisibility statement towards a two-variable main conjecture in the context of nearly ordinary deformations of Galois representations. The obtained results have been incorporated into the paper entitled, ‘Big Heegner point Kolyvagin system for a family of modular forms’ (preprint_1) and has been submitted for publication.
2. Extended Selmer groups, height pairings and Rubin’s formula (Section 1.1.2 of Annex 1): As Dr Buyukboduk has managed to completely describe the relation of Nekovar's abstract height pairings on his extended Selmer groups to p-adic L-functions, in the particular case of the multiplicative group G_m. He indeed carried out the programme outlined in his proposal in order to prove that the cyclotomic units (respectively, the elliptic units) paired under Nekovar's height pairing with certain canonical elements obtained from the Coleman map compute the leading coefficient of an imprimitive Kubota-Leopoldt p-adic L-function (respectively, a certain branch of Katz’s two variable p-adic L-function). This confirms with the general philosophy that Selmer complexes (rather than Selmer groups) should be the correct counterparts of p-adic L-functions.
These results obtained thus far on this problem have been published in Commentarii Mathematici Helvetici, a prestigious mathematics journal.
In addition, Dr Buyukboduk has studied Nekovar's heights for an elliptic curve E which has split multiplicative reduction at p, so as to express the first (respectively second) derivative of the Mazur-Tate-Teitelbaum p-adic L-function in terms of Nekovar's heights when the analytic rank of E is zero (respectively, one). A draft which contains the proof of these very interesting results is available as attached to this report (preprint_5).
3. Kolyvagin systems of rank r (Section 1.1.3 of Annex 1): Dr Buyukboduk successfully developed a machinery of Euler-Kolyvagin systems of rank r for an arbitrary geometric Galois representation T that satisfies a certain Panchishkin condition, generalising the work of Kato, Rubin and Perrin-Riou. This machinery produces a bound on the size of the classical Selmer group attached to T in terms of a certain r × r determinant, a bound that remarkably goes hand in hand with Bloch-Kato conjectures. As one particular example, Dr Buyukboduk utilised an Euler system of rank r whose existence relies on the conjectures of Perrin-Riou on p-adic L-functions. Using his machinery, Dr Buyukboduk produces bounds for the size of the Bloch-Kato Selmer group in terms of L-values using this conjectural Euler system.
The results obtained thus far have been published in Indiana University Mathematics Journal. Further relevant results, concerning the several variable main conjectures for CM fields has been obtained as well, please see the attachment preprint_3.
4. p-adic vs. complex Stark conjectures (Section 1.1.4 of Annex 1). Dr Buyukboduk has obtained a link between the Rubin-Stark elements and Stickelberger elements. This is quite striking as the existence of Rubin-Stark elements is highly conjectural, whereas the existence of Stickelberger elements is unconditional. He has written an article entitled 'Stickelberger elements and Kolyvagin systems' to report on his results which has been published in Nagoya Mathematics Journal.
5. Jochnowitz style congruences (Section 1.1.5 of Annex 1). Dr Buyukboduk has obtained an important general result on the deformations of Kolyvagin systems, which in turn shows that Kato's Kolyvagin systems for modular forms interpolate in families, even beyond the well-studied ordinary case. This significant result has been incorporated in the article entitled 'Deformations of Kolyvagin systems' (preprint_2) and has been submitted for publication.

Final results and potential impact

The results Dr Buyukboduk established during IWTHEGAR will be an important contribution to some of the mainstream problems in the very active research area of number theory. Visits by leading researchers to Istanbul, funded through IWTHEGAR, have given the somewhat isolated mathematical community a unique opportunity to gain familiarity with the mathematical problems at its frontiers. Two of his graduate students have completed their Master's degrees and, Dr Buyukboduk hopes, have obtained the necessary background to pursue PhD degrees. He is currently advising three more graduate students, and he hopes the improved amount of high-quality research output and a stimulating research environment will attract many good students to his area of research. He has built ties with some of the leading researchers in this area, which he hopes to benefit from even after the completion of the project.

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