---
tag: YSAZVCZAZC
kind: concept
name: Nome
msc: ["33E05", 11F27]
sources:
  - url: https://en.wikipedia.org/wiki/Nome_(mathematics)
same_as:
  wikidata: Q7048497
  wikipedia: Nome (mathematics)
---

## Summary

The nome is the variable $q = e^{\pi i\tau}$ in which [[theta functions|2F5CYQJVAF]] and elliptic functions are expanded as power series.
For a real modulus $0 < k < 1$ it is $q = \exp(-\pi K'/K)$, where $K$ and $K'$ are [[complete elliptic integrals of the first kind|RM54CPPE12]].
It ties the modulus of the Jacobi elliptic functions to the period ratio of their lattice.

## Explanations

### L1 human

Elliptic functions depend on a shape parameter, which can be given as the modulus $k$ or as the ratio $\tau$ of two periods.
The nome $q = e^{\pi i\tau}$ is a third way to give the same parameter, chosen so that series expansions are power series in $q$.
For moderate $k$, $q$ is very small, about $k^2/16$ when $k$ is small, so these series converge very fast.
For example, at $k = 1/\sqrt2$ the nome is $e^{-\pi} \approx 0.0432$.