






For a given field $K$ $${\mathbf{P}}^{2}\left(K\right)={K}^{2}\cup \phantom{\rule{0.6em}{0ex}}\left(\text{line at infinity}\right)$$ where $$\begin{array}{cccc}\text{}& \hfill \text{line at infinity}& \hfill =\hfill & \phantom{\rule{0.6em}{0ex}}\left\{\text{classes of parallel lines}\right\}\phantom{\rule{0.6em}{0ex}}\text{in}\phantom{\rule{0.6em}{0ex}}{K}^{2}\hfill \\ \text{}& \hfill \phantom{\rule{0.3em}{0ex}}& \hfill =\hfill & \phantom{\rule{0.6em}{0ex}}\left\{\text{lines through}\phantom{\rule{0.6em}{0ex}}\left(0,0\right)\right\}\phantom{\rule{0.6em}{0ex}}\text{in}\phantom{\rule{0.6em}{0ex}}{K}^{2}\hfill \\ \text{}& \hfill \phantom{\rule{0.3em}{0ex}}& \hfill =\hfill & \phantom{\rule{0.6em}{0ex}}\left\{\text{slopes of lines}\right\}\phantom{\rule{0.6em}{0ex}}\cup \phantom{\rule{0.6em}{0ex}}\left(\infty \right)\hfill \\ \text{}& \hfill \phantom{\rule{0.3em}{0ex}}& \hfill =\hfill & \phantom{\rule{0.6em}{0ex}}K\phantom{\rule{0.3em}{0ex}}\cup \phantom{\rule{0.6em}{0ex}}\left(\infty \right)\hfill \end{array}$$
Each line contains one and only one point (its parallel class) on the line at infinity. The “distinguished point at infinity” is the parallel class of vertical lines.

Given a nonsingular cubic curve $C$, $${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}\phantom{\rule{0.6em}{0ex}}\text{,}$$ with coefficients in $K$, the distinguished point at infinity in ${\mathbf{P}}^{2}\left(K\right)$ lies on $C$.
Proof. Introduce homogeneous coordinates $\left(x,y,z\right)\ne \left(0,0,0\right)$ in ${\mathbf{P}}^{2}$ where:
$\left({x}_{1},{y}_{1},{z}_{1}\right)\equiv \left({x}_{2},{y}_{2},{z}_{2}\right)$ if and only if $\left({x}_{2},{y}_{2},{z}_{2}\right)=t\left({x}_{1},{y}_{1},{z}_{1}\right)$ for some scalar $t\ne 0$.
$\left(x,y,1\right)$ is a homogeneous triple for the affine point $\left(x,y\right)$.
$\left(x,y,z\right)$ is a homogenous triple for an affine point when $z\ne 0$.
$\left(x,y,z\right)$ represents a point on the line at infinity if $z=0$.
$\left(1,m,0\right)$ represents “slope” $m$ on the line at infinity.
$\left(0,1,0\right)$ represents the “distinguished point at infinity”.
In homogeneous coordinates the curve $C$ has the equation $${y}^{2}z+{a}_{1}xyz+{a}_{3}y{z}^{2}={x}^{3}+{a}_{2}{x}^{2}z+{a}_{4}x{z}^{2}+{a}_{6}{z}^{3}\phantom{\rule{0.6em}{0ex}}\text{.}$$
In homogeneous coordinates the line at infinity has the equation $z=0$.
The intersection of the line at infinity with $C$ has the equation ${x}^{3}=0$. Thus, $C$ meets the line at infinity “triply” in the distinguished point at infinity.














Since the mid 20th century one has known that a cusp form $f$ of weight $2$ for ${\Gamma}_{0}\left(N\right)$, not also residing at a level dividing $N$, that is a simultaneous eigenform for the Hecke operators gives rise to an elliptic curve ${E}_{f}$ defined over $\mathbf{Q}$ with conductor $N$.
The Lfunction of ${E}_{f}$ is the Dirichlet series ${\phi}_{f}\left(s\right)$ associated with $f$.
The “Modular Curve Conjecture”, which originated in the mid 20th century, is that every elliptic curve defined over $\mathbf{Q}$ is isogenous to one of those obtained from such a cusp form. (Isogenous elliptic curves share Lfunctions.)
In the mid 1980s it was shown using the theory of representations of the Galois group of the field of all algebraic numbers that the dictionary for elliptic curves defined over $\mathbf{Q}$ provided by the modular curve conjecture (and the extensive knowledge of modular forms) could not possibly contain the FreyHellegouarch curve.
In the 1990s the “Modular Curve Conjecture” was proved.
Fermat's Last Theorem is a corollary of that above.
