XXIX Workshop on Geometric Methods in Physics 27.06-03.07.2010
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# Elena Bunkova

## Universal elliptic formal group law.

Many classical works are dedicated to formal group laws over elliptic curves. The formal group laws play an important role in algebraic topology and in the theory of integrable systems. In the present work we consider elliptic curves, given in Weierstrass parametrisation by the equation
$y^2 + \mu_1 x y + \mu_3 y = x^3 + \mu_2 x^2 + \mu_4 x + \mu_6.$
In Tate coordinates the geometrical addition laws on this curves correspond to the universal formal group law over the ring $\mathbb{Z}[\mu_1, \mu_2, \mu_3, \mu_4, \mu_6]$. We study the structure of this law. In the focus of our interest are the questions, important from the point of view of the problem of integrality of Hirzebruch genera in algebraic topology and the methods of Lax pair in the theory of integrable systems.