Nome (mathematics)

In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by

${\displaystyle q=\mathrm {e} ^{-{\frac {\pi K'}{K}}}=\mathrm {e} ^{\frac {{\rm {i}}\pi \omega _{2}}{\omega _{1}}}=\mathrm {e} ^{{\rm {i}}\pi \tau }\,}$

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 when ${\displaystyle 0<q<1}$. That is, when ${\displaystyle 0<q<1}$, 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. For general ${\displaystyle q\in \mathbb {C} }$ with ${\displaystyle 0<|q|<1}$, ${\displaystyle \tau }$ is not a single-valued function of ${\displaystyle q}$. Explicit expressions for the quarter periods, in terms of the nome, are given in the linked article.

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 ${\displaystyle k}$: ${\displaystyle q=\mathrm {e} ^{-\pi K'(k)/K(k)}}$.

The complementary nome q₁ is given by

${\displaystyle q_{1}=\mathrm {e} ^{-{\frac {\pi K}{K'}}}.\,}$

Sometimes the notation ${\displaystyle q=\mathrm {e} ^{{2{\rm {i}}}\pi \tau }}$ is used for the square of the nome.

Applications[]

The nome solves the following equation:

${\displaystyle k={\frac {\theta _{2}^{2}(x)}{\theta _{3}^{2}(x)}}\rightarrow x=q=\mathrm {e} ^{-\pi K'(k)/K(k)}}$

where ${\displaystyle \theta _{2},\theta _{3}}$ are the Jacobi theta functions and ${\displaystyle K(k)}$ is the complete elliptic integral of the first kind with modulus ${\displaystyle k}$.

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. Some functions that are periodic on the upper half-plane 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 or the square of the nome (the q-expansion) is famously connected to the Fisher-Griess monster by means of monstrous moonshine.

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)^{[citation needed]} 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