Research Project: Iwasawa Theory of Galois Representations (IwTheGaR)
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Contributors
Funders
ID
EC.00007
Authors
Büyükboduk, Kazım
Faculty Member
Publications
Big Heegner point Kolyvagin system for a family of modular forms
(Springer, 2014) Büyükboduk, Kazım; Department of Mathematics; Yes; College of Sciences
The principal goal of this paper is to develop Kolyvagin's descent to apply with the big Heegner point Euler system constructed by Howard for the big Galois representation attached to a Hida family of elliptic modular forms. In order to achieve this, we interpolate and control the Tamagawa factors attached to each member of the family at bad primes, which should be of independent interest. Using this, we then work out the Kolyvagin descent on the big Heegner point Euler system so as to obtain a big Kolyvagin system that interpolates the collection of Kolyvagin systems obtained by Fouquet for each member of the family individually. This construction has standard applications to Iwasawa theory, which we record at the end.
On euler systems of rank r and their Kolyvagin systems
(Indiana University Mathematics Journal, 2010) Büyükboduk, Kazım; Department of Mathematics; Yes; College of Sciences
In this paper we set up a general Kolyvagin system machinery for Euler systems of rank r (in the sense of Perrin-Riou) associated to a large class of Galois representations, building on our previous work on Kolyvagin systems of Rubin-Stark units and generalizing the results of Kato, Rubin and Perrin-Riou. Our machinery produces a bound on the size of the classical Selmer group attached to a Galois representation T (that satisfies certain technical hypotheses) in terms of a certain r x r determinant; a bound which remarkably goes hand in hand with Bloch-Kato conjectures. At the end, we present an application based on a conjecture of Perrin-Riou on p-adic L-functions, which lends further evidence to Bloch-Kato conjectures.
Λ-adic Kolyvagin Systems
(Oxford University Press (OUP), 2011) Büyükboduk, Kazım; Department of Physics; Yes; College of Sciences
In this paper we study Kolyvagin Systems, as defined by Mazur and Rubin, over the cyclotomic Zp-tower for a Gal(Q=Q) representation T. We prove, under certain hypotheses, that the module of -adic Kolyvagin Systems for the cyclotomic deformation T is free of rank one over the cyclotomic Iwasawa algebra. We link our result with a web of conjectures due to Perrin-Riou and Rubin; and we relate the -adic Kolyvagin Systems we prove to exist to (conjectural) p-adic L-functions. We also study the Iwasawa theory of Rubin-Stark elements via the perspective offered by our main theorem, and outline a strategy to deduce the main conjectures of Iwasawa theory for totally real number fields assuming the Rubin-Stark conjecture.
On the Iwasawa theory of CM fields for supersingular primes
(American Mathematical Society (AMS), 2018) Büyükboduk, Kazım; Department of Mathematics; Yes; College of Sciences
The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field F without assuming the p-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark L-restricted Kolyvagin systems, which is constructed out of the conjectural Rubin-Stark Euler system of rank g. (We are also able to obtain weaker unconditional results in this direction.) The second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve E defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for E.
Stark units and the main conjectures for totally real fields
(Foundation Compositio Mathematica, 2009) Büyükboduk, Kazım; Department of Mathematics; Yes; College of Sciences
The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed in an earlier paper also by the author, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this indeed is true. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory, assuming the truth of the Rubin-Stark conjecture and Leopoldt’s conjecture. This statement of local Iwasawa theory turns out to be interesting in its own right as it suggests a relation between solutions to p-adic and complex Stark conjectures.
