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

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

2010-11-07

arxiv.org/abs/1011.1643v3

Mathematics

Classical Analysis and ODEs

Major changes + new results; completely rewritten; 45 pages, no figures, LaTeX

Scientific paper

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 $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (ODEs). Proposed (Hamiltonian) dynamical systems define fundamental differential properties of theta-functions and yield an exponential quadratic extension of the canonical $\theta$-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 technique to the Hitchin case of the sixth Painlev\'e equation.

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