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arXiv · 1011.1643

Non-canonical extension of theta-functions and modular integrability of theta-constants

Yurii V. Brezhnev
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Abstract

This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $θ$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta-functions; they also yield an exponential quadratic extension of the canonical $θ$-series. An integrability condition of these \odes\ explains appearance of the modular $\vartheta$-constants and differential properties thereof. General solutions to all the \odes\ are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences. As a nontrivial application, we apply proposed techni\-que to the Hitchin case of the sixth Painlevé equation.

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Yurii V. Brezhnev. 2013-12-18. Non-canonical extension of theta-functions and modular integrability of theta-constants. https://doi.org/10.1017/s0308210512001023

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