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.

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 $n=1,2$.

• 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 $x^n+y^n+z^n=0$.

• 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 $p\ge 5$.

Let $p\ge 5$ be prime.

Suppose there are non-zero integers $a,b,c$ such that $$a^p+b^p=c^p\text{.}$$

Then the plane cubic curve $$y^2=x\left(x-a^p\right)\left(x+b^p\right)\text{.}$$ is an elliptic curve “defined over” $\mathbf{Q}$ — the Frey-Hellegouarch curve.

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” $$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{,}$$ and over an algebraically closed field of characteristic $\ne 2,3$ into an equation of the form $$y^2=\left(x-{\lambda }_1\right)\left(x-{\lambda }_2\right)\left(x-{\lambda }_3\right)\text{.}$$

The latter is a non-singular cubic when $$\Delta ={\left(\prod _{i<j}\left({\lambda }_i-{\lambda }_j\right)\right)}^2\ne 0\text{.}$$

Example: For the Frey-Hellegouarch curve $$\Delta ={\left(abc\right)}^{2p}\text{.}$$

Given a non-singular cubic curve $C$, $$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{,}$$ 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.

Given a non-singular cubic curve $C$, $$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{,}$$ 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 $K$.

Given a non-singular cubic curve $C$, $$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{,}$$ 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

1. The group origin $0$ is the distinguished point at infinity.

2. 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.

For a given point $\left(c,d\right)$ on the cubic curve $$y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6\text{,}$$ its negative in the group law is the point $\left(c,d^{\prime }\right)$ where $d,d^{\prime }$ are the two roots of $$y^2+\left(a_{1}c+a_3\right)y=c^3+a_2{c}^2+a_{4}c+a_6\text{,}$$ as a quadratic equation in $y$.

• 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.

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 $$N=\prod _{\ell |\Delta }\ell \text{,}$$ 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 $N$.

One defines the “L-series” of $E$ by forming the Euler product, indexed by primes $\ell$ as follows: $$L\left(E,s\right)=\prod _{\ell |N}\frac{1}{1-c_{\ell }{\ell }^{-s}}\prod _{\ell \nmid N}\frac{1}{1-c_{\ell }{\ell }^{-s}+{\ell }^{1-2s}}$$

Expanding the product, one obtains a Dirichlet series $$L\left(E,s\right)=\sum _{k=1}^{\infty }\frac{c_k}{k^s}\text{,}$$ which converges for $\mathrm{Re}\left(s\right)>3⁄2$

Series of this type have been seen in other contexts.

Let $\mathbf{H}$ be $$\mathbf{H}=\left\{\tau \in \mathbf{C}|\mathrm{Im}\left(\tau \right)>0\right\}\text{.}$$

The group $G={\mathrm{SL}}_2\left(\mathbf{R}\right)$ operates via $$M·\tau =\frac{a\tau +b}{c\tau +d},M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right),a,b,c,d\in \mathbf{R},ad-bc=1$$

$G⁄\left\{\pm 1\right\}$ is the group of isometries (distance-preserving analytic maps) of $\mathbf{H}$ relative to the Poincaré metric $$d{s}^2=\frac{d{x}^2+d{y}^2}{y^2}\text{,}\text{for}\tau =x+iy\in H\text{.}$$

(This is the connection with “hyperbolic geometry”.)

Let $G_w$ denote the Eisenstein series $$G_w\left(\tau \right)=\text{const}·\sum _{\left(m,n\right)\in {\mathbf{Z}}^2-\left\{\left(0,0\right)\right\}}\frac{1}{{\left(m\tau +n\right)}^w}\text{,}$$ which converges normally for all $\tau \in \mathbf{H},w\ge 4$.

$G_w\left(\tau \right)$ is not identically $0$ for even $w\ge 4$, while it is self-cancelling for odd $w$.

For given $\tau$ with $g_4\left(\tau \right)=60G_4\left(\tau \right)$, $g_6\left(\tau \right)=140G_6\left(\tau \right)$ the equation $$y^2=4x^3-g_4\left(\tau \right)x-g_6\left(\tau \right)$$ 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 $$C_{{\tau }^{\prime }}\cong C_{\tau }\iff {\tau }^{\prime }=M·\tau \text{for}M\in {\mathrm{SL}}_2\left(\mathbf{Z}\right)\text{.}$$

Thus, over $\mathbf{C}$ $$\left\{\text{isomorphism classes of elliptic curves}\right\}\cong \mathbf{H}⁄{\mathrm{SL}}_2\left(\mathbf{Z}\right)$$

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 $$M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in \Gamma ,\tau \in \mathbf{H}\text{.}$$ 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).

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.

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”.

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 $$\lambda \left(\tau \right)=g_4{\left(\tau \right)}^3-27g_6{\left(\tau \right)}^2$$ is a cusp form of weight $12$ — the smallest weight of a cusp form for ${\Gamma }_0\left(1\right)$.

Let $N\ge 1$ be a positive integer. The group ${\Gamma }_0\left(N\right)$ is given by $$\left\{M=\left(\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right)\in {\mathrm{SL}}_2\left(\mathbf{Z}\right)|c\equiv 0\left(modN\right)\right\}\text{.}$$ In particular $$M_1=\left(\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 0& \hfill 1\end{array}\right)\in {\Gamma }_0\left(N\right)\text{for all}N\ge 1\text{.}$$ 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 $$f\left(\tau \right)=\sum _{k\in \mathbf{Z}}{c}_k{e}^{2\pi ik\tau }\text{.}$$ 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 $$f\left(\tau \right)=\sum _{k=1}^{\infty }{c}_k{e}^{2\pi ik\tau }\text{.}$$ $N$ is called the level.

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 $\mathbf{Q}$: $${\phi }_f\left(s\right)=\prod _{\ell |N}\frac{1}{1-c_{\ell }{\ell }^{-s}}\prod _{\ell \nmid N}\frac{1}{1-c_{\ell }{\ell }^{-s}+{\ell }^{1-2s}}$$

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 $f$.

• $11$ 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 $10000$.

G. Cornell, J. Silverman, & G. Stevens,

Modular Forms and Fermat's Last Theorem,

Springer, 1997

— the record of an instructional conference held at Boston University in August, 1995

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