# Elliptic function of level 4

• E. Yu. Bunkova
Article

## Abstract

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u 2v 2)/(uB(v) − vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u 2 A(v) − v 2 A(u))/(uA(v)2vA(u)2), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u 2 A(v) − v 2 A(u))/(uB(v) − vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A 1, and B″(0) = 2B 2 the following relation holds: (2B(u) + 3A 1 u)2 = 4A(u)3 − (3A 1 2 − 8B 2)u 2 A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.

## References

1. 1.
V. M. Bukhshtaber, “Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups, ” Usp. Mat. Nauk 45 (3), 185–186 (1990) [Russ. Math. Surv. 45 (3), 213–215 (1990)].
2. 2.
V. M. Buchstaber and E. Yu. Bunkova, “Krichever formal groups, ” Funkts. Anal. Prilozh. 45 (2), 23–44 (2011) [Funct. Anal. Appl. 45, 99–116 (2011)].
3. 3.
V. M. Buchstaber and E. Yu. Bunkova, “The universal formal group that defines the elliptic function of level 3, ” Chebyshev. Sb. 16 (2), 66–78 (2015).Google Scholar
4. 4.
V. M. Buchstaber and E. Yu. Bunkova, “Manifolds of solutions for Hirzebruch functional equations, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 136–148 (2015) [Proc. Steklov Inst. Math. 290, 125–137 (2015)].
5. 5.
V. M. Buchstaber and E. Yu. Netay, “CP(2)-multiplicative Hirzebruch genera and elliptic cohomology, ” Usp. Mat. Nauk 69 (4), 181–182 (2014) [Russ. Math. Surv. 69, 757–759 (2014)].
6. 6.
V. M. Buchstaber and I. V. Netay, “Hirzebruch functional equation and elliptic functions of level d, ” Funkts. Anal. Prilozh. 49 (4), 1–17 (2015) [Funct. Anal. Appl. 49, 239–252 (2015)].
7. 7.
V. M. Buchstaber and T. E. Panov, ToricTopology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.Google Scholar
8. 8.
V. M. Buchstaber and A. V. Ustinov, “Coefficient rings of formal groups, ” Mat. Sb. 206 (11), 19–60 (2015) [Sb. Math. 206, 1524–1563 (2015)].
9. 9.
E. Yu. Bunkova, V. M. Buchstaber, and A. V. Ustinov, “Coefficient rings of Tate formal groups determining Krichever genera, ” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 292, 43–68 (2016) [Proc. Steklov Inst. Math. 292, 37–62 (2016)].
10. 10.
11. 11.
F. Hirzebruch, “Elliptic genera of level N for complex manifolds, ” Preprint 88-24 (Max-Planck-Inst. Math., Bonn, 1988).Google Scholar
12. 12.
F. Hirzebruch, T. Berger, and R. Jung, Manifoldsand Modular Forms (Vieweg, Braunschweig, 1992), Aspects Math. E20.
13. 13.
I. M. Krichever, “Generalized elliptic genera and Baker–Akhiezer functions, ” Mat. Zametki 47 (2), 34–45 (1990) [Math. Notes 47, 132–142 (1990)].
14. 14.
S. Lang, EllipticFunctions (Springer, New York, 1987).Google Scholar
15. 15.
S. Ochanine, “Sur les genres multiplicatifs définis par des intégrales elliptiques, ” Topology 26 (2), 143–151 (1987).
16. 16.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Part II: The Transcendental Functions, Repr. of the 4th ed. 1927 (Cambridge Univ. Press, Cambridge, 1996).