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

Abstract

We present new results in the theory of the classical theta 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 the appearance of the modular ϑ-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.

Critical points of the classical Eisenstein series of weight two

“…Thenτ = 1 2 + i 4b ∈ {τ ∈ H| Re τ = 1 2 }, namelyτ is the unique zero of η 1 on the line {τ ∈ H| Re τ = 1 2 } with b 0 := Imτ ∈ ( 5 24 , 1 2 √ 3 ). Recall (3.5) and (4.1) that η 1…”

“…We will prove in Section 2 that for each C ∈ R \ {0, 1}, there is a unique τ(C) ∈ F 0 such that (1.4) holds. 1 Consequently, the parametrization (1.4) will give three smooth curves C 0 := {τ(C)|C ∈ (0, 1)}, C − := {τ(C)|C ∈ (−∞, 0)}, C + := {τ(C)|C ∈ (1, +∞)}.…”

“…The relation between (1.4) and η 1 (τ) comes from the classical formula (see e.g. [1], or from Ramanujan's formula: E 2 (τ) = πi 6 (E 2 2 − E 4 )) (1.5) η 1 (τ) = i 2π η 1 (τ) 2 − 1 12 g 2 (τ) . 1 Note that the RHS of (1.4) is actually a multi-valued function, please see Theorem 3.1…”

On a Quantization of the Classical θ-Functions

“…One may be noted in this connection that complete quantization procedure of theta functions will perhaps inevitably involve, the holomorphic integrals excepted, the full set of Abelian objects on Abelian variety: meromorphic functions, meromorphic and logarithmic integrals, and, which is the most nontrivial point, the θ itself as integral of an Abelian meromorphic integral. Such a nature of the theta-function is explained in [5].…”

“…Complete differential properties of Jacobian series were described very recently [5] and we reproduce them in a nutshell. These four θ-series are defined as [1,18] …”