In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by
\( q=e^{{-{\frac {\pi K'}{K}}}}=e^{{{\frac {{{\rm {{i}}}}\pi \omega _{2}}{\omega _{1}}}}}=e^{{{{\rm {{i}}}}\pi \tau }}\, \)
where K and iK′ are the quarter periods, and ω1 and ω2 are the fundamental pair of periods, and τ = iK ′/K = ω2/ω1 is the half-period ratio. The nome can be taken to be a function of any one of these quantities; conversely, any one of these quantities can be taken as functions of the nome. Each of them uniquely determines the others. That is, the mappings between these various symbols are both 1-to-1 and onto, and so can be inverted: the quarter periods, the half-periods and the half-period ratio can be explicitly written as functions of the nome. Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article. Conversely, the above can be taken as an explicit expression for the nome, in terms of the other quantities.
Thus, the nome can be take to be either a function, or as a parameter; conversely, the quarter and half periods can be taken either as functions, or as parameters; specifying any one is sufficient to uniquely determine all of the others; they are all functions of one-another.
Notationally, the quarter periods K and iK′ are usually used only in the context of the Jacobian elliptic functions, whereas the half-periods ω1 and ω2 are usually used only in the context of Weierstrass elliptic functions. Some authors, notably Apostol, use ω1 and ω2 to denote whole periods rather than half-periods.
The nome is frequently used as a value with which elliptic functions and modular forms can be described; on the other hand, it can also be thought of as function, because the quarter periods are functions of the elliptic modulus. This ambiguity occurs because for real values of the elliptic modulus, the quarter periods and thus the nome are uniquely determined.
The complementary nome q1 is given by
\( q_{1}=e^{{-{\frac {\pi K}{K'}}}}.\, \)
Some sources however use convention \( {\displaystyle q=e^{{2{\rm {i}}}\pi \tau }} \) or \( {\displaystyle q=e^{{2{\rm {i}}}\pi z}} \) .
See the articles on quarter period and elliptic integrals for additional definitions and relations on the nome.
Applications
The nome is commonly used as the starting point for the construction of Lambert series, the q-series and more generally the q-analogs. That is, the half-period ratio τ is commonly used as a coordinate on the complex upper half-plane, typically endowed with the Poincaré metric to obtain the Poincaré half-plane model. The nome then serves as a coordinate on a punctured disk of unit radius; it is punctured because q=0 is not part of the disk (or rather, q=0 corresponds to τ → ∞). This endows the punctured disk with the Poincaré metric.
The upper half-plane (and the Poincaré disk, and the punctured disk) can thus be tiled with the fundamental domain, which is the region of values of the half-period ratio τ (or of q, or of K and iK′ etc.) that uniquely determine a tiling of the plane by parallelograms. The tiling is referred to as the modular symmetry given by the modular group. Functions that are periodic on the upper half-plane (or periodic on the Poincaré disk or periodic on the punctured q-disk) are called to as modular functions; the nome, the half-periods, the quarter-periods or the half-period ratio all provide different parameterizations for these periodic functions.
The prototypical modular function is Klein's j-invariant. It can be written as a function of either the half-period ratio τ or as a function of the nome q. The series expansion in terms of the nome (the q-expansion) is famously connected the Fisher-Griess monster by means of monstrous moonshine.
Functions that are "almost periodic", but not quite, and having a particular transformation under the modular group are called modular forms. For example, Euler's function arises as the prototype for q-series in general.
The nome, as the q of q-series then arises in the theory of affine Lie algebras, essentially because (to put it poetically, but not factually) those algebras describe the symmetries and isometries of Riemann surfaces.
References
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. OCLC 1097832 . See sections 16.27.4 and 17.3.17. 1972 edition: ISBN 0-486-61272-4
Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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