Comparison of classifications of 2-dimensional local fields, type II
DOI:
https://doi.org/10.21638/spbu01.2020.404Abstract
The article contributes to the theory of elimination of wild ramification for 2-dimensional local fields. It continues the study of classification of complete discrete valuation fields introduced in the work of Masato Kurihara. The main object of study is a 2-dimensional local field K of mixed characteristic with a finite residue field of odd characteristic. If such a field is weakly unramified over its constant subfield k (the maximal usual local field inside it), i. e., if eK/k = 1, its structure is well known. It is also known that any 2-dimensional local field can be turned into a standard one by means of a finite extension of its constant subfield (Epp’s theorem). However, the estimate of the minimal degree of such extension is an open question in general. In Kurihara’s article the 2-dimensional are subdivided into 2 types as follows. Consider a non-trivial linear relation between differentials of the two local parameters of the field. The field belongs to Type I, if the valuation of the coefficient by the uniformizer is less than that by the second local parameter, and to Type II otherwise. This paper is devoted to the fields of Type II. For them we consider the invariant Δ, the difference between valuations of coefficients in the above mentioned linear relation (it is non-positive for the fields of Type II). The minimal degree of the required extension of k cannot be less than eK/k for trivial reasons. However, such extension of degree eK/k does not exist in general. In this article it is proved that it exists if the absolute value of Δ is sufficiently large. The corresponding estimate for Δ depends only on eK/k.Keywords:
higher local fields, wild ramification
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Литература
1. Иванова О. Ю. О связи классификации Курихары с теориейустранения ветвления // Алгебра и анализ. 2012. Т. 24, №2. С. 130–153.
2. Иванова О. Ю. Классификация Курихары и расширения максимальнойглубины для многомерных локальных полей// Алгебра и анализ. 2012. Т. 24, №6. С. 42–76.
3. Ivanova O., Vostokov S., Zhukov I. On two approaches to classification of higher local fields // Чебышёвскийсборник. 2019. Т. 20. Вып. 2. С. 177–189.
4. Востоков С. В., Жуков И. Б., Иванова О. Ю. Инварианты Курихары и устранение высшего ветвления // Зап. научн. семин. ПОМИ. 2020. В печати.
5. Epp H. Eliminating wild ramification // Invent. Math. 1973. Vol. 19. P. 235–249.
6. Жуков И. Б., Коротеев М. В. Устранение высшего ветвления // Алгебра и анализ. 1999. Т. 11, №6. С. 153–177.
7. Kurihara M. On two types of complete discrete valuation fields // Compos. Math. 1987. Vol. 63. P. 237–257.
8. Kurihara M. Two types of complete discrete valuation fields. In: Geometry & Topology Monographs. 2000. Vol. 3. Invitation to higher local fields. P. 109–112. https://doi.org/10.2140/gtm.2000.3.109
9. Hyodo O. Wild ramification in the imperfect residue field case. In: Adv. Stud. Pure Math. 1987. Vol. 12. Galois Representations and Arithmetic Algebraic Geometry. P. 287–314. https://doi.org/10.2969/aspm/01210287
10. Жуков И. Б., Мадунц А. И. Многомерные полные поля: топология и другие основные понятия // Тр. С.-Петерб. мат. общ-ва. 1995. Т. 3. С. 4–46.
11. Zhukov I. Higher dimensional local fields. In: Geometry & Topology Monographs. 2000. Vol. 3. Invitation to higher local fields. P. 5–18. https://doi.org/10.2140/gtm.2000.3.5
12. Жуков И. Б., Мадунц А. И. Аддитивные и мультипликативные разложения в многомерных локальных полях // Зап. научн. семин. ПОМИ. 2000. Т. 272. С. 186–196.
References
1. Ivanova O. Yu., “On connection of Kurihara’s classification with the theory of elimination of ramification”, St. Petersburg Math. J. 24(2), 283–299 (2013).
2. Ivanova O. Yu., “Kurihara classification and maximal depth extensions for multidimensional local fields”, St. Petersburg Math. J. 24(6), 877–901 (2012).
3. Ivanova O., Vostokov S., Zhukov I., “On two approaches to classification of higher local fields”, Chebyshevskii Sbornik 20(2), 177–189 (2019).
4. Vostokov S. V., Zhukov I. B., Ivanova O. Yu., “Kurihara’s invariants and elimination of wild ramification”, Zapiski Nauchnykh Seminarov POMI, in press.
5. Epp H., “Eliminating wild ramification”, Invent. Math. 19, 235–249 (1973).
6. Zhukov I. B., Koroteev M.V., “Elimination of wild ramification”, St. Petersburg Math. J. 11(6), 1063–1083 (2000).
7. Kurihara M., “On two types of complete discrete valuation fields”, Compos. Math. 63, 237–257 (1987).
8. Kurihara M., “Two types of complete discrete valuation fields”, in Geometry & Topology Monographs, vol. 3. Invitation to higher local fields, 109–112 (2000). https://doi.org/10.2140/gtm.2000.3.109
9. Hyodo O., “Wild ramification in the imperfect residue field case”, in: Adv. Stud. Pure Math., vol. 12. Galois Representations and Arithmetic Algebraic Geometry, 287–314 (1987). https://doi.org/10.2969/aspm/01210287
10. Zhukov I. B., Madunts A. I., “Multidimensial complete fields: topology and other basic constructions”, Proceedings of the St. Petersburg Mathematical Society 3, 4–46 (1995).
11. Zhukov I., “Higher dimensional local fields” in: Geometry & Topology Monographs, vol. 3. Invitation to higher local fields, 5–18 (2000). https://doi.org/10.2140/gtm.2000.3.5
12. Zhukov I. B., Madunts A. I., “Additive and multiplicative decompositions in multidimensional local fields”, J. Math. Sci. (N. Y.) 116(1), 2987–2992 (2003).
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Published
2020-12-27
How to Cite
Ivanova, O. Y., & Zhukov, I. B. (2020). Comparison of classifications of 2-dimensional local fields, type II. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(4), 607–621. https://doi.org/10.21638/spbu01.2020.404
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On the anniversary of S. V. Vostokov
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
