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On Addition Theorems Related to Elliptic Integrals

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Abstract

We present formulas for the components of the Buchstaber formal group law and its exponent over ℚ[p1, p2, p3, p4]. This leads to an addition theorem for the general elliptic integral \(\int_0^x {dt{\rm{/}}R\left( t \right)} \) with \(R(t)=\sqrt{1+p_{1} t+p_{2} t^{2}+p_{3} t^{3}+p_{4} t^{4}}\). The study is motivated by Euler’s addition theorem for elliptic integrals of the first kind.

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References

  1. M. Bakuradze, “The formal group laws of Buchstaber, Krichever, and Nadiradze coincide,” Russ. Math. Surv. 68(3), 571–573 (2013) [transl. from Usp. Mat. Nauk 68 (3), 189–190 (2013)].

    Article  Google Scholar 

  2. M. Bakuradze, “On the Buchstaber formal group law and some related genera,” Proc. Steklov Inst. Math. 286, 1–15 (2014) [transl. from Tr. Mat. Inst. Steklova 286, 7–21 (2014)].

    Article  MathSciNet  Google Scholar 

  3. M. Bakuradze, “Computing the Krichever genus,” J. Homotopy Relat. Struct. 9(1), 85–93 (2014).

    Article  MathSciNet  Google Scholar 

  4. V. M. Buhštaber, “The Chern–Dold character in cobordisms. I,” Math. USSR, Sb. 12(4), 573–594 (1970) [transl. from Mat. Sb. 83 (4), 575–595 (1970)].

    Article  Google Scholar 

  5. V. M. Bukhshtaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups,” Russ. Math. Surv. 45(3), 213–215 (1990) [transl. from Usp. Mat. Nauk 45 (3), 185–186 (1990)].

    Article  MathSciNet  Google Scholar 

  6. V. M. Buchstaber, “Complex cobordism and formal groups,” Russ. Math. Surv. 67(5), 891–950 (2012) [transl. from Usp. Mat. Nauk 67 (5), 111–174 (2012)].

    Article  MathSciNet  Google Scholar 

  7. V. M. Buchstaber and E. Yu. Bunkova, “Krichever formal groups,” Funct. Anal. Appl. 45(2), 99–116 (2011) [transl. from Funkts. Anal. Prilozh. 45 (2), 23–44 (2011)].

    Article  MathSciNet  Google Scholar 

  8. V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.

    Book  Google Scholar 

  9. V. M. Buchstaber and A. V. Ustinov, “Coefficient rings of formal group laws,” Sb. Math. 206(11), 1524–1563 (2015) [transl. from Mat. Sb. 206 (11), 19–60 (2015)].

    Article  MathSciNet  Google Scholar 

  10. G. Höhn, “Komplexe elliptische Geschlechter und S 1-äquivariante Kobordismustheorie,” Diplomarbeit (Bonn Univ., Bonn, 1991); arXiv: math/0405232 [math.AT].

    Google Scholar 

  11. I. M. Krichever, “Generalized elliptic genera and Baker–Akhiezer functions,” Math. Notes 47(2), 132–142 (1990) [transl. from Mat. Zametki 47 (2), 34–45 (1990)].

    Article  MathSciNet  Google Scholar 

  12. S. P. Novikov, “The methods of algebraic topology from the viewpoint of cobordism theory,” Math. USSR, Izv. 1(4), 827–913 (1967) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 31 (4), 855–951 (1967)].

    Article  Google Scholar 

  13. S. Ochanine, “Sur les genres multiplicatifs définis par des intégrales elliptiques,” Topology 26(2), 143–151 (1987).

    Article  MathSciNet  Google Scholar 

  14. D. Quillen, “On the formal group laws of unoriented and complex cobordism theory,” Bull. Am. Math. Soc. 75, 1293–1298 (1969).

    Article  MathSciNet  Google Scholar 

  15. S. Schreieder, “Dualization invariance and a new complex elliptic genus,” J. Reine Angew. Math. 692, 77–108 (2014); arXiv: 1109.5394v3 [math.AT].

    MathSciNet  MATH  Google Scholar 

  16. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (Univ. Press, Cambridge, 1927).

    MATH  Google Scholar 

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Funding

The work was supported by the CNRS PICS, grant no. 7736. The first author was also supported by the Shota Rustaveli National Science Foundation of Georgia, grant no. 217-614.

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Authors and Affiliations

  1. Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, Chavchavadze Ave. 1, 0179, Tbilisi, Georgia

    Malkhaz Bakuradze

  2. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Case courrier 051, Place Eugène Bataillon, 34090, Montpellier, France

    Vladimir V. Vershinin

  3. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia

    Vladimir V. Vershinin

Authors
  1. Malkhaz Bakuradze
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  2. Vladimir V. Vershinin
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Corresponding authors

Correspondence to Malkhaz Bakuradze or Vladimir V. Vershinin.

Additional information

This article was submitted by the authors simultaneously in Russian and English

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 305, pp. 29–39.

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Bakuradze, M., Vershinin, V.V. On Addition Theorems Related to Elliptic Integrals. Proc. Steklov Inst. Math. 305, 22–32 (2019). https://doi.org/10.1134/S0081543819030027

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  • DOI: https://doi.org/10.1134/S0081543819030027

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