## Appendix A Elliptic Curves  
Equation (11) is a particular case of equations of the form:  
\[k^{2}+a_{1}k\omega+a_{3}k=\omega^{3}+a_{2}\omega^{2}+a_{4}\omega+a_{6} \tag{12}\]  
which are called _Weierstrass equations_[53]. The curves described by these equations may have interesting forms, and some examples are shown in Figure 11.  
As we can see in this figure, these curves may have singular points where the curve intersects with itself (as in the curve \(k^{2}=\omega^{3}-3\omega+2\) shown in Figure A1a), or where the curve presents cusps (as in the curve \(k^{2}=\omega^{3}\), shown in Figure A1b), but they may also describe regular curves without intersections or cusps (as in the curve \(k^{2}=4\omega^{3}+4\omega\) shown in Figure A1c). Weierstrass equations without singular points are termed _regular_ (_nonsingular_) _Weierstrass equations_, and the curves described by these equations are so important in some contexts that they have received a special name: _elliptic curves_. Therefore, we have the following:  
**Definition A1**. : _An elliptic curve is a curve described by a regular Weierstrass equation._  
This definition implies that every elliptic curve is described by a Weierstrass equation. The converse, however, is not true: not every equation of the form (A1) defines an elliptic curve. Only the _regular_ Weistrass equations describe elliptic curves. Now, as the rescaled dispersion relation (11) is a Weierstrass equation, we would like to know if it describes an elliptic curve. In other words, we would like to know if the Weierstrass equation (11) is a _regular_ one, or not. The answer to this query can be easily found, as there is a well-known criterion which tells us when a Weierstrass equation is _regular_[53]. This criterion can be easily modified to tell us when an equation of the form:  
\[k^{2}+a_{1}k\omega+a_{3}k=a_{0}\omega^{3}+a_{2}\omega^{2}+a_{4}\omega+a_{6}\] (A2)  
describes a _regular_ curve without singular points (notice that (A2) only differs from (A1) by the presence of the coefficient \(a_{0}\)). The criterion is the following:  
**Criterion A1**. : _An equation of the form (A2) is regular if and only if the discriminant \(\Delta\) defined below is not zero._  
To define the discriminant \(\Delta\) it is useful to define the following quantities:  
\[b_{2}=\Big{(}a_{1}^{2}+4a_{2}\Big{)}/a_{0}\] (A3)  
\[b_{4}=\left(2a_{4}+a_{1}a_{3}\right)/a_{0}\] (A4)  
\[b_{6}=\Big{(}a_{3}^{2}+4a_{6}\Big{)}/a_{0}\] (A5)  
\[b_{8}=\Big{(}a_{1}^{2}a_{6}+4a_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^{ 2}\Big{)}/a_{0}^{2}\] (A6)  
In terms of these quantities we can write the discriminant \(\Delta\) in the following form:  
\[\Delta=-b_{2}^{2}b_{8}-8b_{4}^{3}-27b_{6}^{2}+9b_{2}b_{4}b_{6}\] (A7)  
If we put \(a_{0}=1\) in Equations (A3)-(A6), Criterion A1 tells us when a Weierstrass equation is regular. If we apply this criterion to the Weierstrass equations which describe the curves shown in Figure A1 we will see that only the curve (c) is an elliptic curve, since the discriminant \(\Delta\) is zero for curves (a) and (b). Now we would like to determine if our dispersion relation describes an elliptic curve. Equation (A2) reduces to Equation (10) if the coefficients \(a_{n}\) take the values:  
\[a_{1}=a_{4}=a_{6}=0,\ a_{0}=c_{3}/c_{0},\ a_{2}=-c_{2}/c_{0},\ a_{3}=1/c_{0},\] (A8)  
and in this case Equation (A7) takes the simpler form:  
\[\Delta=16c_{2}^{2}/\Big{(}c_{0}c_{3}^{4}\Big{)}\] (A9)Therefore, if \(c_{0}\), \(c_{2}\) and \(c_{3}\) are different from zero, the condition \(\Delta\
eq 0\) will be satisfied, thus implying that the dispersion relation (10) is always a regular curve, and its rescaled form (11) is always an elliptic curve. At this point it may be worth observing that if we apply in Equation (A2) the following changes of variables:  
Figure 11.0 (A10)  
Equation (A2) transforms into:  
Figure 11.1 (A11)  
where \(\alpha_{2}=\left(a_{2}+a_{1}^{2}/4\right)/a_{0}^{2/3}\), \(\alpha_{4}=\left(a_{4}-a_{1}a_{3}/2\right)/a_{0}^{1/3}\) and \(\alpha_{6}=a_{3}+a_{3}^{2}/4\).