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20 Theta FunctionsApplications

§20.12 Mathematical Applications


§20.12(i) Number Theory

For applications of θ3(0,q) to problems involving sums of squares of integers see §27.13(iv), and for extensions see Estermann (1959), Serre (1973, pp. 106–109), Koblitz (1993, pp. 176–177), and McKean and Moll (1999, pp. 142–143).

For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s τ(n) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). For an application of a generalization in affine root systems see Macdonald (1972).

§20.12(ii) Uniformization and Embedding of Complex Tori

For the terminology and notation see McKean and Moll (1999, pp. 48–53).

The space of complex tori /(+τ) (that is, the set of complex numbers z in which two of these numbers z1 and z2 are regarded as equivalent if there exist integers m,n such that z1-z2=m+τn) is mapped into the projective space P3 via the identification z(θ1(2z|τ),θ2(2z|τ),θ3(2z|τ),θ4(2z|τ)). Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)).