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Büyükboduk, Kazım

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Kazım

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Büyükboduk

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Büyükboduk, Kazım

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2009 - 200912010 - 201815

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Researcher Of Publications: search.filters.isAuthorOfPublication.57e13e6d-21b6-417c-997b-1848ab4ce754

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    On the anticyclotomic Iwasawa theory of CM forms at supersingular primes
    (European Mathematical Soc, 2015) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    In this paper, we study the anticyclotomic Iwasawa theory of a CM form f of even weight w >= 2 at a supersingular prime, generalizing the results in weight 2, due to Agboola and Howard. In due course, we are naturally lead to a conjecture on universal norms that generalizes a theorem of Perrin-Riou and Berger and another that generalizes a conjecture of Rubin (the latter seems linked to the local divisibility of Heegner points). Assuming the truth of these conjectures, we establish a formula for the variation of the sizes of the Selmer groups attached to the central critical twist of f as one climbs up the anticyclotomic tower. We also prove a statement which may be regarded as a form of the anticyclotomic main conjecture (without p-adic L-functions) for the central critical twist of f.
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    Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
    (De Gruyter, 2018) Lei, Antonio; Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Z(p)-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.
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    On euler systems of rank r and their Kolyvagin systems
    (Indiana Univ Math Journal, 2010) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    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.
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    Big Heegner point Kolyvagin system for a family of modular forms
    (Springer International Publishing Ag, 2014) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    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.
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    On the Iwasawa theory of cm fields for supersingular primes
    (American Mathematical Society (AMS), 2018) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; 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.
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    P-adic variation in arithmetic geometry: a survey
    (Springer, 2017) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences
    The main goal of this survey is to provide a general overview of the theme of p-adic variation, both from a historical and technical view point. We start off with Kummer’s work and Iwasawa’s treatment of cyclotomic fields, which eventually paved the way to the modern p-adic variational techniques. These methods have proved extremely powerful and enabled us to gain access to some of the most important problems in mathematics, such as the Bloch-Kato conjectures and Langlands’ Programme. We will point at a variety of concrete applications in this vein.
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    Coleman-adapted Rubin–Stark Kolyvagin systems and supersingular Iwasawa theory of CM abelian varieties
    (Wiley, 2015) Lei, Antonio; Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    The goal of this article was to study the Iwasawa theory of an abelian variety A that has complex multiplication by a complex multiplication (CM) field F that contains the reflex field of A, which has supersingular reduction at every prime above p. To do so, we make use of the signed Coleman maps constructed in our companion article [Kazim Buyukboduk and Antonio Lei, 'Integral Iwasawa theory of motives for non-ordinary primes', 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed p-adic L-function via a reciprocity conjecture that we formulate for the (conjectural) Rubin-Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank-g EUler-Kolyvagin systems to be applied with Rubin-Stark elements and deduce the main conjecture for the maximal Z(p)-power extension of F for the primes failing the ordinary hypothesis of Katz.
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    Integral Iwasawa theory of galois representations for non-ordinary primes
    (Springer, 2017) Lei, Antonio; Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are 0 or 1 (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define integral p-adic L-functions and (one unconditionally and other conjecturally) cotorsion Selmer groups. This allows us to reformulate Perrin-Riou's main conjecture in terms of these objects, in the same fashion as Kobayashi's +/--Iwasawa theory for supersingular elliptic curves. By the aid of the theory of Coleman-adapted Kolyvagin systems we develop here, we deduce parts of Perrin-Riou's main conjecture from an explicit reciprocity conjecture.
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    Deformations of Kolyvagin systems
    (Springer, 2016) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Ochiai has previously proved that the Beilinson–Kato Euler systems for modular forms interpolate in nearly-ordinary p-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai’s work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson–Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus), assuming that the deformation problem at hand is unobstructed in the sense of Mazur. We then use what we call universal Kolyvagin systems to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a p-adic L-function (in a certain sense of the word, in 3-variables).
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    On Nekovar's heights, exceptional zeros and a conjecture of Mazur-Tate-Teitelbaum
    (Oxford University Press (OUP), 2016) Department of Mathematics; Büyükboduk, Kazım; Faculty Member; Department of Mathematics; College of Sciences; N/A
    Let E/ℚ be an elliptic curve which has split multiplicative reduction at a prime p and whose analytic rank ran(E) equals one. The main goal of this article is to relate the second-order derivative of the Mazur-Tate-Teitelbaum p-adic L-function Lp(E,s) of E to Nekovář's height pairing evaluated on natural elements arising from the Beilinson-Kato elements. Along the way, we extend a Rubin-style formula of Nekovář to apply in the presence of exceptional zeros. Our height formula allows us, among other things, to compare the order of vanishing of Lp(E,s) at s=1 with its (complex) analytic rank ran(E) assuming the non-triviality of the height pairing. This has consequences toward a conjecture of Mazur, Tate, and Teitelbaum.