### 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 language | English |
---|---|

Pages (from-to) | 59-65 |

Number of pages | 7 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Scopus subject areas

- Mathematics(all)

### Cite this

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**Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field.** / Madunts, A. I.; Vostokov, S. V.; Vostokova, R. P.

Research output

TY - JOUR

T1 - Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

AU - Madunts, A. I.

AU - Vostokov, S. V.

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

Y1 - 2019/1/1

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.

KW - Lubin—Tate formal group

KW - convergence of power series

KW - multidimensional local field

KW - multidimensional local field

KW - LubinTate formal group

KW - convergence of power series

UR - http://www.scopus.com/inward/record.url?scp=85064895600&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/formal-groups-subrings-ring-integers-multidimensional-local-field/

U2 - 10.3103/S1063454119010084

DO - 10.3103/S1063454119010084

M3 - Article

AN - SCOPUS:85064895600

VL - 52

SP - 59

EP - 65

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -