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E. Yu. Bunkova 《Proceedings of the Steklov Institute of Mathematics》2016,294(1):201-214
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) = (u2 ? v2)/(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) = (u2A(v) ? v2A(u))/(uA(v)2 ? vA(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) = (u2A(v) ? v2A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A1, and B″(0) = 2B2 the following relation holds: (2B(u) + 3A1u)2 = 4A(u)3 ? (3A12 ? 8B2)u2A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions. 相似文献
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E. Yu. Bunkova V. M. Buchstaber 《Proceedings of the Steklov Institute of Mathematics》2009,266(1):1-28
In the framework of S.P. Novikov’s program for boosting the effectiveness of thetafunction formulas of finite-gap integration
theory, a system of differential equations for the parameters of the sigma function in genus 2 is constructed. A counterpart
of this system in genus 1 is equivalent to the Chazy equation. On the basis of the obtained results, a two-dimensional analog
of the Frobenius-Stickelberger connection is defined and calculated. 相似文献
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Mathematical Notes - We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic... 相似文献
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On the basis of the general Weierstrass model of the cubic curve with parameters μ = (μ1, μ2, μ3, μ4, μ6), the explicit form of the formal group that corresponds to the Tate uniformization of this curve is described. This formal
group is called the general elliptic formal group. The differential equation for its exponential is introduced and studied.
As a consequence, results on the elliptic Hirzebruch genus with values in ℤ[μ] are obtained. 相似文献
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We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations. 相似文献
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Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a... 相似文献
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Elena Yu. Bunkova 《Proceedings of the Steklov Institute of Mathematics》2018,302(1):33-47
The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χa,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?k is a series with parameters in the Zariski closure of U in ?k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?Pn?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?4. 相似文献
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Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g > 0$$ , a... 相似文献
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