Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

Research output

Abstract

We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

Original languageEnglish
Pages (from-to)59-65
Number of pages7
JournalVestnik St. Petersburg University: Mathematics
Volume52
Issue number1
DOIs
Publication statusPublished - 1 Jan 2019

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Formal Group
Local Field
Ring
Integer
Isogeny
Coefficient
Power series
Homomorphism
Explicit Formula
Maximal Ideal
Zero
Endomorphisms
One to one correspondence

Scopus subject areas

  • Mathematics(all)

Cite this

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title = "Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field",
abstract = "We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.",
keywords = "Lubin—Tate formal group, convergence of power series, multidimensional local field, multidimensional local field, LubinTate formal group, convergence of power series",
author = "Madunts, {A. I.} and Vostokov, {S. V.} and Vostokova, {R. P.}",
note = "Madunts, A. I., Vostokov, S. V., & Vostokova, R. P. (2019). Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field. Vestnik St. Petersburg University: Mathematics, 52(1), 59–65. https://doi.org/10.3103/S1063454119010084",
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AU - Vostokova, R. P.

N1 - Madunts, A. I., Vostokov, S. V., & Vostokova, R. P. (2019). Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field. Vestnik St. Petersburg University: Mathematics, 52(1), 59–65. https://doi.org/10.3103/S1063454119010084

PY - 2019/1/1

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N2 - We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

AB - We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

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