## Appendix B [PERSON]'s Last Theorem  
The fact that the linear dispersion relation of Equation (4) is an elliptic curve suggests that this equation might be related to other subjects where these curves also play an important role. Among these subjects there is one which stands out among the others: [PERSON]'s last theorem. The proof of this theorem rests, precisely, on the existence of a one-to-one correspondence between elliptic curves and a rare kind of complex functions: _modular forms_. Every elliptic curve is related to a modular form, and consequently, as Equation (4) is related to an elliptic curve, it should also bear some relationship with a modular form. Moreover, also the nonlinear extensions of Equation (4), such as the following:  
\[iu_{z}+c_{0}u_{zz}+c_{2}u_{tt}-ic_{3}u_{ttt}+\gamma_{1}|u|^{2}u=0\] (A12)  
\[iu_{z}+c_{0}u_{zz}+c_{2}u_{tt}-ic_{3}u_{ttt}+\gamma_{1}|u|^{2}u-\gamma_{2}|u|^{ 4}u=0\] (A13)  
might be related to modular forms, since the three equations (Equations (4), (A12) and (A13)) share the same linear dispersion relation (the elliptic curve (10)). Therefore, as it might be important to advance in the study of the relationships between Equations (4), (A12) and (A13), elliptic curves and modular forms, in the following we describe the main steps which lead to the proof of [PERSON]'s last theorem.  
STEP 1: The [PERSON] conjecture.  
In 1955 the young mathematician [PERSON] discovered that an unexpected (and surprising) relation seemed to exist between elliptic curves and _modular forms_. These \"forms\" are complex functions \(F(\tau)\) which satisfy (among other requisites) the following condition [55]:  
\[F\bigg{(}\frac{a\tau+b}{c\tau+d}\bigg{)}=\left(c\tau+d\right)^{k}F(\tau)\] (A14)  
where \(a\), \(b\), \(c\), \(d\) and \(k\) are integers such that \(ad-bc=1\), and \(\tau\) is a complex number with positive imaginary part. [PERSON], and a friend of him, [PERSON], studied various elliptic curves, and all of them turned out to be related to modular forms. Therefore, they conjectured that a one-to-one correspondence existed between elliptic curves and modular forms.  
STEP 2: [PERSON]'s discovery.  
In 1975 [PERSON] discovered that the elliptic curve [56]:  
\[y^{2}=x(x-a^{p})(x-b^{p}) \tag{100}\]  
where \(a\), \(b\) and \(p>2\) are integers, would have rather unusual properties if \(a^{p}+b^{p}\) were also a p_th_ power, i.e., if there existed an integer \(c\) such that:  
\[a^{p}+b^{p}=c^{p} \tag{101}\]  
STEP 3: [PERSON]'s proposition.  
In 1984 [PERSON] announced in a conference in Oberwolfach, Germany, that if there existed integers \(a\), \(b\), \(c\) and \(p>2\) satisfying (101), then the elliptic curve (100) would not have a modular form associated with it [57]. In other words: if [PERSON]'s last theorem were false, then the Taniyama-Shimura would also be false, and this implied that:  
If the Taniyama-Shimura is true, then [PERSON]'s last theorem is also true!  
Therefore, [PERSON] had discovered that one way to prove [PERSON]'s last theorem was to prove the Taniyama-Shimura conjecture. However, a problem remained: [PERSON]'s procedure to \"prove\" that the elliptic curve (100) had not a corresponding modular form, contained an error. This problem was solved in 1986 by [PERSON], who presented a correct proof of [PERSON]'s proposition [57]. Therefore, the way to prove [PERSON]'s last theorem was clearly indicated: it was necessary to prove the [PERSON] conjecture.  
STEP 4: [PERSON] proof.  
In 1993 [PERSON] announced that he had proved a restricted form of the Taniyama-Shimura conjecture that was sufficient to prove [PERSON]'s last theorem. It turned out that [PERSON] proof had a mistake, but in 1994, in collaboration with [PERSON], the mistake was corrected, and [PERSON]'s last theorem was finally proved. The full Taniyama-Shimura conjecture (now known as _modularity theorem_) was proved in 2001 by [PERSON], [PERSON], [PERSON] and [PERSON], thus completing the intellectual adventure started by [PERSON] and [PERSON] 46 years before [58].