A q-Analogue of Mahler Expansions, I

Conrad, Keith

doi:10.1006/aima.1999.1890

Cited by 19 publications

(15 citation statements)

References 18 publications

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“…2 is the image of 2Z 2 under the isomorphism given by 3 k . To obtain a q-Mahler basis as in [20] with q = 9 it is important that ν 2 (9 − 1) > 0. The q-Mahler basis is a basis for numerical polynomials with domain restricted to 2Z 2 .…”

Section: The Cooperations Of Ko and Bomentioning

confidence: 99%

On the ring of cooperations for 2‐primary connective topological modular forms

Behrens

Ormsby

Stapleton

et al. 2019

Journal of Topology

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We analyze the ring tmf * tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E2-term of the Adams spectral sequence for tmf ∧ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf * tmf in TMF * TMF Q admits a description in terms of 2-variable modular forms, and (3) modulo v2-torsion, tmf * tmf injects into a certain product of copies of π * TMF0(N ), for various values of N . We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf * tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ∧ tmf gives a connective cover of TMF0(3), and show that another piece gives a connective cover of TMF0(5). To help motivate our methods, we also review the existing work on bo * bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.for bo ∧ bo (respectively, tmf ∧ tmf) splits as a direct sum of Ext-groups for the integral (respectively, bo) Brown-Gitler spectra. Section 2.4 recalls some exact sequences used in [9] which allow for an inductive approach for computing Ext of bo-Brown-Gitler comodules, and introduces related sequences which allow for an inductive approach to Ext groups of integral Brown-Gitler comodules. Section 3This section is devoted to the motivating example of bo ∧ bo. Sections 3.1-3.3 are primarily expository, based upon the foundational work of Adams, Lellmann, Mahowald, and Milgram. We make an effort to consolidate their theorems and recast them in modern notation and terminology, and hope that this will prove a useful resource to those trying to learn the classical theory of bo-cooperations and v 1 -periodic stable homotopy. To the best of our knowledge, Sections 3.4 and 3.5 provide new perspectives on this subject. Section 3.1 is devoted to the homology of the HZ i and certain Ext A(1) * -computations relevant to the Adams spectral sequence computation of bo * bo.We shift perspectives in Section 3.2 and recall Adams's description of KU * KU in terms of numerical polynomials. This allows us to study the image of bu * bu in KU * KU as a warm-up for our study of the image of bo * bo in KO * KO.We undertake this latter study in Section 3.3, where we ultimately describe a basis of KO 0 bo in terms of the '9-Mahler basis' for 2-adic numerical polynomials with domain 2Z 2 . By studying the Adams filtration of this basis, we are able to use the above results to fully describe bo * bo mod v 1 -torsion elements.In Section 3.4, we link the above two perspectives, studying the image of bo * HZ i in KO * KO. Theorem 3.6 provides a complete description of this image (mod v 1 -torsion) in terms of the 9-Mahler basis.We conclude with Section 3.5 which studies a certain mapconstructed from Adams operations. We show that this map is an injection after applying π * and exhibit how it interacts with the Brown...

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Section: The Cooperations Of Ko and Bomentioning

confidence: 99%

On the ring of cooperations for 2‐primary connective topological modular forms

Behrens

Ormsby

Stapleton

et al. 2019

Journal of Topology

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“…It is now well known (cf. [1] K. Conrad) that the sequence C n n≥0 is an orthonormal basis of p K : this means that any element f ∈ p K can be written as a convergent sum f = n≥0 a n C n , a n ∈ K, lim n→+ a n = 0 and f = sup n≥0 a n .…”

Section: Preliminariesmentioning

confidence: 97%

Ap-Adic Quantum Weyl Algebra

Tangara

2010

Communications in Algebra

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“…However, if we restrict q = 1, we have x n 1 = x n . We now state the q-analogue of Mahler's theorem given by Conrad [6].…”

Section: Preliminaries On the Q-mahler Theoremmentioning

confidence: 98%

Characterization of the ergodicity of 1-Lipschitz functions on Z2 using the q-Mahler basis

Jeong

2015

Journal of Number Theory

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The ergodicity of 1-Lipschitz functions on Z 2 represented by the Mahler basis was characterized by V.S. Anashin (1994) in [1]. His results are mainly based on the so-called folklore criterion for ergodicity, depending on the algebraic normal form of Boolean functions associated with coordinate functions. In this paper, we employ the q-Mahler basis to provide q-analogues of Anashin's results whose proof does not rely on this criterion.

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A q-Analogue of Mahler Expansions, I

Cited by 19 publications

References 18 publications

On the ring of cooperations for 2‐primary connective topological modular forms

On the ring of cooperations for 2‐primary connective topological modular forms

Ap-Adic Quantum Weyl Algebra

Characterization of the ergodicity of 1-Lipschitz functions on Z2 using the q-Mahler basis

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