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Elliptic Invariants

The invariants of a Weierstrass elliptic function P(z|omega_1,omega_2) are defined by the Eisenstein series

g_2(omega_1,omega_2)=60sum^'_(m,n)Omega_(mn)^(-4)
(1)
g_3(omega_1,omega_2)=140sum^'_(m,n)Omega_(mn)^(-6).
(2)

Here,

Omega_(mn)(omega_1,omega_2)=2momega_1+2nomega_2,
(3)

where omega_1 and omega_2 are the half-periods of the elliptic function. The Wolfram Language command WeierstrassInvariants[{omega1, omega2}] gives the invariants g_2 and g_3 corresponding to the half-periods omega_1 and omega_2.

Writing g_i(tau)=g_i(1,tau),

g_2(tau)=g_2(1,tau)=omega_1^4(omega_1,omega_2)
(4)
g_3(tau)=g_3(1,tau)=omega_1^6(omega_1,omega_2),
(5)

and the invariants have the Fourier series

g_2(tau)=(4pi^4)/3[1+240sum_(k=1)^(infty)sigma_3(k)e^(2piiktau)]
(6)
g_3(tau)=(8pi^6)/(27)[1-504sum_(k=1)^(infty)sigma_5(k)e^(2piiktau)]
(7)

where tau=omega_1/omega_2 is the half-period ratio and sigma_k(n) is the divisor function (Apostol 1997).

See also

Dedekind Eta Function, Eisenstein Series, Modular Discriminant, Tau Function, Weierstrass Elliptic Function

Related Wolfram sites

http://functions.wolfram.com/EllipticFunctions/WeierstrassInvariants/

Explore with Wolfram|Alpha

References

Apostol, T. M. "The Fourier Expansions of g_2(tau) and g_3(tau)." §1.9 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 12-13, 1997.Brezhnev, Y. V. "Uniformisation: On the Burnside Curve y^2=x^5-x." 9 Dec 2001. http://arxiv.org/abs/math.CA/0111150.

Referenced on Wolfram|Alpha

Elliptic Invariants

Cite this as:

Weisstein, Eric W. "Elliptic Invariants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipticInvariants.html

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