On Higher Special Elements of<i>p</i>-adic Representations

Burns, David J.; Sano, Takamichi; Tsoi, Kwok-Wing

doi:10.1093/imrn/rnz378

Cited by 6 publications

(9 citation statements)

References 34 publications

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“…Remark 3.17. (a) The existence of pairings of the displayed shape in Theorem 3.16(2) was first observed (at least in the case of representations with coefficients in Gorenstein orders in finite-dimensional Q-algebras) by Burns, Sano and Tsoi [BST19]. The above result can therefore be seen as a natural Iwasawa-theoretic analogue of the pairing constructed for T by the aforementioned authors.…”

Section: An Iwasawa-theoretic Pairingmentioning

confidence: 70%

On universal norms for $p$-adic representations in higher-rank Iwasawa theory

Bullach

Daoud

2021

Acta Arith.

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The investigation of the deep connection between Lfunctions and arithmetic is at the heart of modern number theory. By now we have a number of partial results on this matter, many of which due to celebrated results obtained via the Euler system method that has been developed by Thaine, Kolyvagin, Rubin and Mazur.However, all of these (unconditional) results are restricted to cases where the order of vanishing of the L-function is at most one. Although a notion of higher-rank Euler system was already established by Perrin-Riou more than 20 years ago, technical issues arising from the use of exterior powers hindered the theory surrounding higher-rank Euler systems from being fully operational. These technical obstructions have only recently been overcome by Burns, Sakamoto and Sano in a series of articles ([BS21], [BSS19a],[BSS19b] and [BSS19c]). Key to their approach is the consistent use of exterior biduals instead of exterior powers, a notion that is based on the lattice introduced by Rubin [Rub96, §1.2] and provides better functorial properties in many aspects.Since Euler systems are, by their very definition, universal norms on Z pextensions, we feel that the study of higher-rank universal norms undertaken in this article naturally fits into the chain of developments described above. As in the aforementioned works, the use of exterior biduals allows us to develop a theory that naturally extends the classical theory of universal norms to both the higher-rank and equivariant settings.Overview of results. To explain our results in a little more detail, we first introduce some notation. Let L|K be a finite abelian extension of

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Section: An Iwasawa-theoretic Pairingmentioning

confidence: 70%

On universal norms for $p$-adic representations in higher-rank Iwasawa theory

Bullach

Daoud

2021

Acta Arith.

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“…Proof. This follows directly from the argument of [11,Th. 3.3.7] after fixing the data (R, C, J, a, a ′ ) in loc.…”

Section: 4mentioning

confidence: 87%

“…in which the first isomorphism is induced by (11) and the isomorphism E 1 (Q p ) ⊗ Zp Q p ≃ Q p induced by the logarithm log ω associated to the fixed Néron differential ω, and the last isomorphism is the obvious identification. Any choice of basis element x of the Z-module r Z E(Q) tf therefore gives rise to an isomorphism of Q p -spaces…”

Section: Review Of the Generalized Perrin-riou Conjecturementioning

confidence: 99%

On derivatives of Kato's Euler system and the Mazur-Tate Conjecture

Burns¹,

Kurihara²,

Sano³

2021

Preprint

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We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank. Contents 1. Introduction 1.1. Discussion of the main results 1.2. General notation 2. Review of the Mazur-Tate Conjecture 2.1. The Birch and Swinnerton-Dyer Conjecture 2.2. Modular elements and the Mazur-Tate Conjecture 3. Review of the Generalized Perrin-Riou Conjecture 3.1. The Bockstein regulator 3.2. Kato's zeta elements 3.3. The Generalized Perrin-Riou Conjecture 4. The main result 4.1. Statement of the main result and the deduction of Theorem 1.1 4.2. Determinantal zeta elements 5. The proof of Theorem 4.6 and the deduction of Corollary 1.2 5.1. Construction of the modular element 5.2. Bloch-Kato Selmer complexes 5.3. Bockstein maps for Selmer complexes 5.4. The algebraic Birch and Swinnerton-Dyer element 5.5. Completion of the proof of Theorem 4.6 5.6. The deduction of Corollary 1.2 References

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“…Non-abelian higher special elements. In the commutative setting Burns, Sano and the second author [14] have recently developed a theory of 'higher special elements' as a generalisation of the notion of higher rank Euler systems. Such elements are also associated to admissible complexes in the sense of [10] and live in the higher exterior powers of the cohomology modules of the complexes.…”

mentioning

confidence: 99%

“…In this way, we hope to contribute to the future study of non-commutative versions of higher rank Euler systems. We emphasise that, exactly as in the commutative case considered in [14], our construction of higher special elements does not depend on fixing 'separable' tuples of elements (in the highest degree non-trivial cohomology modules of admissible complexes) but rather arises from arbitrary choices of tuples. This fact makes our arithmetic applications significantly finer than the similar theories currently present in the literature, as we shall illustrate in the rest of this introduction.…”

mentioning

confidence: 99%

On non-abelian higher special elements of $p$-adic representations

Castillo¹,

Tsoi²

2019

Preprint

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We develop a theory of 'non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of p-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined Stark conjectures and of refined conjectures of Birch and Swinnerton-Dyer type.

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On Higher Special Elements ofp-adic Representations

Cited by 6 publications

References 34 publications

On universal norms for $p$-adic representations in higher-rank Iwasawa theory

On universal norms for $p$-adic representations in higher-rank Iwasawa theory

On derivatives of Kato's Euler system and the Mazur-Tate Conjecture

On non-abelian higher special elements of $p$-adic representations

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