Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 

 

Explicit form of the Hilbert symbol for polynomial formal groups


Authors: S. Vostokov and V. Volkov
Translated by: V. Volkov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 785-796
MSC (2010): Primary 11S31; Secondary 14L05
DOI: https://doi.org/10.1090/spmj/1358
Published electronically: July 27, 2015
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a local field, $ c$ a unit in $ K$, and $ F_c (X,Y) = X + Y + cXY$ a polynomial formal group that gives rise to a formal module $ F_c(\mathfrak{M})$ on the maximal ideal in the ring of integers of $ K$. Assume that $ K$ contains the group $ \mu _{F_c, n}$ of the roots of isogeny $ [p^n]_c(X)$. The natural Hilbert symbol $ (\,\cdot \,,\,\cdot \,)_c \colon K^*\times F_c(\mathfrak{M}) \to \mu _{F_c, n}$ is defined over the module $ F(\mathfrak{M})$. An explicit formula for $ (\,\cdot \,,\,\cdot \,)_c$ is constructed.


References [Enhancements On Off] (What's this?)

  • 1. S. V. Vostokov, An explicit form of the reciprocity law, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1288–1321, 1439 (Russian). MR 522940
  • 2. I. R. Šafarevič, A general reciprocity law, Mat. Sbornik N.S. 26(68) (1950), 113–146 (Russian). MR 0031944
  • 3. I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, 2nd ed., Translations of Mathematical Monographs, vol. 121, American Mathematical Society, Providence, RI, 2002. With a foreword by I. R. Shafarevich. MR 1915966
  • 4. S. V. Vostokov, The Hilbert symbol in a discretely valued field, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 94 (1979), 50–69, 150 (Russian). Rings and modules, 2. MR 571515
  • 5. S. V. Vostokov and E. V. Ferens-Sorotskiy, Hilbert pairing for the polynomial formal groups, Vestnik St. Petersburg Univ. Math. 43 (2010), no. 1, 18–22. MR 2662405, https://doi.org/10.3103/S1063454110010048
  • 6. S. V. Vostokov, V. V. Volkov, and G. K. Pak, The Hilbert symbol for polynomial formal groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 400 (2012), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 23, 127–131, 247 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 192 (2013), no. 2, 196–199. MR 3029567
  • 7. K. Hensel, Die multiplicative Dars ellung der algebraischen Zahlen fur den Bereich eines beliebigen Prin teil, J. Reine Angew. Math. 136 (1916).
  • 8. S. V. Vostokov, On I. R. Shafarevich’s paper “A general reciprocity law”, Mat. Sb. 204 (2013), no. 6, 3–22 (Russian, with Russian summary); English transl., Sb. Math. 204 (2013), no. 5-6, 781–800. MR 3113452
  • 9. S. V. Vostokov, Normed pairing in formal modules, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 765–794, 966 (Russian). MR 548504
  • 10. S. V. Vostokov, Symbols on formal groups, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 5, 985–1014, 1198 (Russian). MR 637613
  • 11. D. G. Benua and S. V. Vostokov, Norm pairing in formal groups and Galois representations, Algebra i Analiz 2 (1990), no. 6, 69–97 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 6, 1221–1249. MR 1092526
  • 12. V. A. Abrashkin, Explicit formulas for the Hilbert symbol of a formal group over Witt vectors, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 3, 3–56 (Russian, with Russian summary); English transl., Izv. Math. 61 (1997), no. 3, 463–515. MR 1478558, https://doi.org/10.1070/im1997v061n03ABEH000122
  • 13. S. V. Vostokov and O. V. Demchenko, Explicit form of the Hilbert pairing for relative formal Lubin-Tate groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 227 (1995), no. Voprosy Teor. Predstav. Algebr i Grupp. 4, 41–44, 156 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 89 (1998), no. 2, 1105–1107. MR 1374555, https://doi.org/10.1007/BF02355855
  • 14. S. V. Vostokov and O. V. Demchenko, An explicit formula for the Hilbert pairing of formal Honda groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 86–128, 346 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 116 (2003), no. 1, 2926–2952. MR 1811794, https://doi.org/10.1023/A:1023494524764

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 11S31, 14L05

Retrieve articles in all journals with MSC (2010): 11S31, 14L05


Additional Information

S. Vostokov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: sergei.vostokov@gmail.com

V. Volkov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: vladvolkov239@gmail.com

DOI: https://doi.org/10.1090/spmj/1358
Keywords: Polynomial formal groups, formal groups, Hilbert symbol, local rings
Received by editor(s): January 10, 2014
Published electronically: July 27, 2015
Additional Notes: Supported by RFBR (grant no. 14-01-00393)
Article copyright: © Copyright 2015 American Mathematical Society