# Abstract

#### Invitation to the Proof of Fermat's Last “Theorem”

 Fermat's Last “Theorem” (ca. 1637) was finally proved in the mid-1990s by using the study of plane cubic curves of the form ${y}^{2}=\left(x-A\right)\left(x-B\right)\left(x-C\right)$ where A, B, and C are distinct integers. This talk will provide an overview of the main ingredients.

# 1.  The Statement

 There is no solution in positive integers $x,y,z$ of the equation ${x}^{n}+{y}^{n}={z}^{n}$ for $n\ge 3$. Note: There are infinitely many essentially different solutions when $n=1,2$.

# 2.  Old History

 The statement is equivalent to the statement that there are no non-zero integers $x,y,z$ satisfying ${x}^{n}+{y}^{n}={z}^{n}$ for $n\ge 3$. The statement is equivalent to the statement that there are no rational points off the coordinate axes on the plane curve ${x}^{n}+{y}^{n}=1$ for $n\ge 3$. For odd exponents $n\ge 3$ the statement is equivalent to the statement that there are no non-zero integers such that ${x}^{n}+{y}^{n}+{z}^{n}=0$.

# 3.  Old History (continued)

 If the theorem is true when the exponent is a given $n$, then it is certainly true when the exponent is a multiple of that value of $n$. The case where $n$ is $3$ or $4$ can be handled within the realm of “elementary” number theory. (See, for example, the classic text of Hardy & Wright.) Any integer $n\ge 3$ not divisible by $4$ must be divisible by an odd prime. It remains to prove the theorem when the exponent $n$ is a prime $p\ge 5$.

# 4.  A Solution leads to a Cubic Curve

 Let $p\ge 5$ be prime. Suppose there are non-zero integers $a,b,c$ such that ${a}^{p}+{b}^{p}={c}^{p}\phantom{\rule{0.6em}{0ex}}\text{.}$ Then the plane cubic curve ${y}^{2}=x\left(x-{a}^{p}\right)\left(x+{b}^{p}\right)\phantom{\rule{0.6em}{0ex}}\text{.}$ is an elliptic curve “defined over” $\mathbf{Q}$ — the Frey-Hellegouarch curve.

# 5.  Cubic Curves

 Over any field $K$, e.g., $\mathbf{Q}$, $\mathbf{C}$, or ${{\mathbb{F}}}_{p}=\mathbf{Z}⁄p\mathbf{Z}$, after a (projective) change of coordinates in $K$ a non-singular cubic curve with at least one $K$-valued point may be brought into “generalized Weierstrass form” ${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}\phantom{\rule{0.6em}{0ex}}\text{,}$ and over an algebraically closed field of characteristic $\ne 2,3$ into an equation of the form ${y}^{2}=\left(x-{\lambda }_{1}\right)\left(x-{\lambda }_{2}\right)\left(x-{\lambda }_{3}\right)\phantom{\rule{0.6em}{0ex}}\text{.}$ The latter is a non-singular cubic when $\Delta ={\left(\prod _{i Example: For the Frey-Hellegouarch curve $\Delta ={\left(abc\right)}^{2p}\phantom{\rule{0.6em}{0ex}}\text{.}$

# 6.  Cubics over the Complex Numbers

 Given a non-singular 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 $\mathbf{C}$ and $\Delta \ne 0$, the set of all solutions $\left(x,y\right)$ in ${\mathbf{C}}^{2}$ together with the “distinguished point at infinity” forms a compact Riemann surface of genus one — a torus.

# 7.  The Projective Plane

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

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.

# 8.  A Line Meets a Cubic in 3 Points

 Given a non-singular 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 $K$, every line in ${K}^{2}$ passing through $2$ points of $C$ meets $C$ in a third point, allowing for multiplicities. Proof. Parameterize the line and get a cubic equation in the parameter with two known roots in $K$.

# 9.  The Distinguished Point at Infinity

Given a non-singular 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.

# 10.  The Group Law

 Given a non-singular 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 $K$, there is a unique “algebraic” group law on the points of $C$ in ${\mathbf{P}}^{2}\left(K\right)$ characterized by the two conditions The group origin $0$ is the distinguished point at infinity. For three points $P,Q,R$ of $C$ one has $P+Q+R=0$ if and only if $P,Q,R$ lie on a line. Note: Although the commutative law is obviously automatic here, it is not easy to check the associative law.

# 11.  The Group Negative

 For a given point $\left(c,d\right)$ on the cubic curve ${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}\phantom{\rule{0.6em}{0ex}}\text{,}$ its negative in the group law is the point $\left(c,{d}^{\prime }\right)$ where $d,{d}^{\prime }$ are the two roots of ${y}^{2}+\left({a}_{1}c+{a}_{3}\right)y={c}^{3}+{a}_{2}{c}^{2}+{a}_{4}c+{a}_{6}\phantom{\rule{0.6em}{0ex}}\text{,}$ as a quadratic equation in $y$.

# 12.  Elliptic Curves

 The non-singular cubic curves defined over $K$ with at least one $K$-valued point are the “group objects” in the category of algebraic curves defined over $K$. For a curve in generalized Weierstrass form, the required $K$-valued point may always be taken to be the distinguished point at infinity. These are called elliptic curves. When $K=\mathbf{Q}$, much is known about them. Modular forms — objects associated with hyperbolic geometry — provide a dictionary for elliptic curves defined over $\mathbf{Q}$. The Frey-Hellegouarch curve cannot be in that dictionary.

# 13.  The mod $\ell$ reduction of an elliptic curve

 Let $E$ be an elliptic curve of the form ${y}^{2}=\left(x-A\right)\left(x-B\right)\left(x-C\right)$ where $A,B,C$ are distinct integers. When $\ell$ is a prime not dividing $\Delta$ (the square of the product of the root differences), $E$ determines also a curve ${E}_{\ell }$ defined over the finite field ${{\mathbb{F}}}_{\ell }=\mathbf{Z}⁄\ell \mathbf{Z}$. ${E}_{\ell }$ is non-singular when $l$ is not a factor of $\Delta$. For our purposes, i.e., in the case of the Frey-Hellegouarch curve, the conductor $N$ of $E$ may be defined to be $N=\prod _{\ell |\Delta }\ell \phantom{\rule{0.6em}{0ex}}\text{,}$ the square-free part of $\Delta$. Let ${c}_{\ell }$ be defined by ${c}_{\ell }=1-\left|E\left({{\mathbb{F}}}_{\ell }\right)\right|+\ell$ when $l\nmid N$. Here $\left|E\left({{\mathbb{F}}}_{\ell }\right)\right|$ denotes the number of points of ${E}_{\ell }$ in the field ${{\mathbb{F}}}_{\ell }$. ${c}_{\ell }$ is defined in a slightly more complicated way for each of the finitely many primes $\ell$ dividing $N$.

# 14.  The L-series of $E$

 One defines the “L-series” of $E$ by forming the Euler product, indexed by primes $\ell$ as follows: $L\left(E,s\right)\phantom{\rule{0.6em}{0ex}}=\phantom{\rule{0.6em}{0ex}}\prod _{\ell |N}\frac{1}{1-{c}_{\ell }{\ell }^{-s}}\phantom{\rule{0.6em}{0ex}}\prod _{\ell \nmid N}\frac{1}{1-{c}_{\ell }{\ell }^{-s}+{\ell }^{1-2s}}$ Expanding the product, one obtains a Dirichlet series $L\left(E,s\right)=\sum _{k=1}^{\infty }\frac{{c}_{k}}{{k}^{s}}\phantom{\rule{0.6em}{0ex}}\text{,}$ which converges for $\mathrm{Re}\left(s\right)>3⁄2$ Series of this type have been seen in other contexts.

# 15.  Isometries of the Upper-Half Plane

 Let $\mathbf{H}$ be $\mathbf{H}=\left\{\tau \in \mathbf{C}\phantom{\rule{0.4em}{0ex}}|\phantom{\rule{0.4em}{0ex}}\mathrm{Im}\left(\tau \right)>0\right\}\phantom{\rule{0.6em}{0ex}}\text{.}$ The group $G={\mathrm{SL}}_{2}\left(\mathbf{R}\right)$ operates via $M·\tau =\frac{a\tau +b}{c\tau +d},\phantom{\rule{1.8em}{0ex}}M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right),\phantom{\rule{0.6em}{0ex}}a,b,c,d\in \mathbf{R},\phantom{\rule{0.6em}{0ex}}ad-bc=1$ $G⁄\left\{±1\right\}$ is the group of isometries (distance-preserving analytic maps) of $\mathbf{H}$ relative to the Poincaré metric $d{s}^{2}=\frac{d{x}^{2}+d{y}^{2}}{{y}^{2}}\phantom{\rule{0.6em}{0ex}}\text{,}\phantom{\rule{0.6em}{0ex}}\text{for}\phantom{\rule{0.6em}{0ex}}\tau =x+iy\in H\phantom{\rule{0.6em}{0ex}}\text{.}$ (This is the connection with “hyperbolic geometry”.)

# 16.  Family of Elliptic Curves over $\mathbf{C}$

 Let ${G}_{w}$ denote the Eisenstein series ${G}_{w}\left(\tau \right)=\text{const}·\sum _{\left(m,n\right)\in {\mathbf{Z}}^{2}-\left\{\left(0,0\right)\right\}}\phantom{\rule{0.6em}{0ex}}\frac{1}{{\left(m\tau +n\right)}^{w}}\phantom{\rule{0.6em}{0ex}}\text{,}$ which converges normally for all $\tau \in \mathbf{H},\phantom{\rule{0.6em}{0ex}}w\ge 4$. ${G}_{w}\left(\tau \right)$ is not identically $0$ for even $w\ge 4$, while it is self-cancelling for odd $w$. For given $\tau$ with ${g}_{4}\left(\tau \right)=60{G}_{4}\left(\tau \right)$, ${g}_{6}\left(\tau \right)=140{G}_{6}\left(\tau \right)$ the equation ${y}^{2}=4{x}^{3}-{g}_{4}\left(\tau \right)x-{g}_{6}\left(\tau \right)$ gives rise to a cubic curve ${C}_{\tau }$ in classical Weierstrass form. Every elliptic curve defined over $\mathbf{C}$ occurs this way, and one has ${C}_{{\tau }^{\prime }}\cong {C}_{\tau }\phantom{\rule{0.6em}{0ex}}⇔\phantom{\rule{0.6em}{0ex}}{\tau }^{\prime }=M·\tau \phantom{\rule{0.6em}{0ex}}\text{for}M\in {\mathrm{SL}}_{2}\left(\mathbf{Z}\right)\phantom{\rule{0.6em}{0ex}}\text{.}$ Thus, over $\mathbf{C}$ $\left\{\text{isomorphism classes of elliptic curves}\right\}\phantom{\rule{0.6em}{0ex}}\cong \mathbf{H}⁄{\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$

# 17.  Modular Forms

 The Eisenstein series are examples of modular forms: complex-valued holomorphic functions $f$ in $\mathbf{H}$ satisfying $f\left(M·\tau \right)={\left(c\tau +d\right)}^{w}f\left(\tau \right)$ for $M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in \Gamma \phantom{\rule{0.3em}{0ex}},\phantom{\rule{1.8em}{0ex}}\tau \in \mathbf{H}\phantom{\rule{0.6em}{0ex}}\text{.}$ where $\Gamma$ is ${\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$ or a subgroup of finite index in ${\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$. The integer $w$ is the weight of $f$. ${G}_{w}$ is a modular form of weight $k$. A modular form is, more or less, a holomorphic section of a “line bundle” on the quotient space $\mathbf{H}⁄\Gamma$. Modular forms are also required to be “holomorphic at cusps”, i.e., approach a finite limit at a “cusp” (see below).

# 18.  Action of ${\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$ on $\mathbf{H}$

The action of $\Gamma ={\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$ on $\mathbf{H}$ is portrayed in this picture:

• The gray area is a fundamental domain. It has infinite extent.

• $\mathbf{H}⁄\Gamma$ is non-compact.

# 19.  Cusps Compactify the Quotient

 Let $\Gamma$ be a subgroup of finite index in ${\Gamma }_{0}\left(1\right)={\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$. $\mathbf{H}⁄\Gamma$ “covers” $\mathbf{H}⁄{\Gamma }_{0}\left(1\right)$ $\Gamma$ operates on ${\mathbf{H}}^{*}=\mathbf{H}\cup \mathbf{Q}\cup \left\{\infty \right\}$. For $\Gamma =\Gamma \left(1\right)$ the orbit of $\infty$ is $\mathbf{Q}\cup \left\{\infty \right\}$. For general $\Gamma$ the number of orbits in $\mathbf{Q}\cup \left\{\infty \right\}$ is finite. ${\mathbf{H}}^{*}⁄\Gamma$ compactifies $\mathbf{H}⁄\Gamma$ by adjoining the finitely many “cusps”.

# 20.  Cusp Forms

 Let $\Gamma$ be a subgroup of finite index in ${\Gamma }_{0}\left(1\right)={\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$. A modular form for $\Gamma$ is a cusp form if its limiting value at each cusp of $\Gamma$ is $0$. Example: For $\Gamma ={\Gamma }_{0}\left(1\right)={\mathrm{SL}}_{2}\left(\mathbf{Z}\right)$ the modular form $\lambda \left(\tau \right)={g}_{4}{\left(\tau \right)}^{3}-27{g}_{6}{\left(\tau \right)}^{2}$ is a cusp form of weight $12$ — the smallest weight of a cusp form for ${\Gamma }_{0}\left(1\right)$.

# 21.  The Groups ${\Gamma }_{0}\left(N\right)$

 Let $N\ge 1$ be a positive integer. The group ${\Gamma }_{0}\left(N\right)$ is given by $\left\{M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in {\mathrm{SL}}_{2}\left(\mathbf{Z}\right)\phantom{\rule{0.4em}{0ex}}|\phantom{\rule{0.4em}{0ex}}c\equiv 0\phantom{\rule{0.3em}{0ex}}\left(modN\right)\right\}\phantom{\rule{0.6em}{0ex}}\text{.}$ In particular ${M}_{1}=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right)\in {\Gamma }_{0}\left(N\right)\phantom{\rule{0.6em}{0ex}}\text{for all}\phantom{\rule{0.6em}{0ex}}N\ge 1\phantom{\rule{0.6em}{0ex}}\text{.}$ If $f$ is a modular form, then $f\left({M}_{1}·\tau \right)=f\left(\tau +1\right)=f\left(\tau \right)$ is periodic, so has a Fourier expansion $f\left(\tau \right)=\sum _{k\in \mathbf{Z}}{c}_{k}{e}^{2\pi ik\tau }\phantom{\rule{0.6em}{0ex}}\text{.}$ Because $f$ is holomorphic at the cusp $\infty$ one has ${c}_{k}=0$ for $k<0$, and if $f$ is a cusp form ${c}_{0}=0$ so that then $\phantom{\rule{0.6em}{0ex}}f\left(\tau \right)=\sum _{k=1}^{\infty }{c}_{k}{e}^{2\pi ik\tau }\phantom{\rule{0.6em}{0ex}}\text{.}$ $N$ is called the level.

# 22.  The Dirichlet Series

 There are certain operators, called Hecke operators ${\left\{{T}_{w}\left(m\right)\right\}}_{m\ge 1}$, that act semi-simply on the space of cusp forms for ${\Gamma }_{0}\left(N\right)$ not coming from levels dividing $N$. The structure of the algebra of these operators shows that if $f\left(\tau \right)=\sum _{k=1}^{\infty }{c}_{k}{e}^{2\pi ik\tau }$ is a cusp form of weight 2 that is a simultaneous eigenform of these operators then the corresponding Dirichlet series ${\phi }_{f}\left(s\right)=\sum _{k=1}^{\infty }\frac{{c}_{k}}{{k}^{s}}$ has an Euler product expansion just like the Euler product that is the L-function of an elliptic curve defined over $\mathbf{Q}$: ${\phi }_{f}\left(s\right)=\phantom{\rule{0.6em}{0ex}}\prod _{\ell |N}\frac{1}{1-{c}_{\ell }{\ell }^{-s}}\phantom{\rule{0.6em}{0ex}}\prod _{\ell \nmid N}\frac{1}{1-{c}_{\ell }{\ell }^{-s}+{\ell }^{1-2s}}$

# 23.  Cusp Forms of Weight $2$ on ${\Gamma }_{0}\left(N\right)$

 A cusp form $f$ of weight $2$ for ${\Gamma }_{0}\left(N\right)$ is essentially a regular differential on the quotient ${X}_{0}\left(N\right)={\mathbf{H}}^{*}⁄{\Gamma }_{0}\left(N\right)$. When $f$, not coming from levels dividing $N$, is an eigenform of the Hecke operators, it determines in a straightforward way a 1-dimensional quotient variety of the Jacobian variety ${J}_{0}\left(N\right)$ of ${X}_{0}\left(N\right)$, $\begin{array}{cccc}\hfill {X}_{0}\left(N\right)\hfill & \hfill \to \hfill & \hfill {J}_{0}\left(N\right)\hfill & \hfill \to {E}_{f}\hfill \end{array}$ which quotient is an elliptic curve ${E}_{f}$ defined over $\mathbf{Q}$ with conductor $N$, and, therefore a regular map – the modular parametrization of ${E}_{f}$ – from ${X}_{0}\left(N\right)$ to ${E}_{f}$ with the property that the unique (up to a constant) regular differential on ${E}_{f}$ pulls back to the differential on ${X}_{0}\left(N\right)$ determined by $f$.

# 24.  The Dictionary for Elliptic Curves over $\mathbf{Q}$

• 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 L-function 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 L-functions.)

• 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 Frey-Hellegouarch curve.

• In the 1990s the “Modular Curve Conjecture” was proved.

• Fermat's Last Theorem is a corollary of that above.

# 25.  Dictionary Trivia

 $11$ is the smallest value of $N$ for which there is a non-zero cusp form of weight $2$ for the group ${\Gamma }_{0}\left(N\right)$. In this case the dimension of the space of cusp forms is $1$. There are 3 non-isomorphic but isogenous elliptic curves with conductor $11$: ${y}^{2}+y={x}^{3}-{x}^{2}-10x-20$ ${y}^{2}+y={x}^{3}-{x}^{2}-7820x-263580$ ${y}^{2}+y={x}^{3}-{x}^{2}$ The Cremona database — an encoding of the dictionary — has been built into Sage (http://www.sagemath.org/). Documentation for its use may be found at http://www.sagemath.org/doc/reference/sage/databases/cremona.html. The smallest conductor having more than $1$ isogeny class is $26$, which has $2$. The smallest conductor having more than $2$ isogeny classes is $57$, which has $3$. There are $38402$ isogeny classes with conductors smaller than $10000$.