Karl Rubin's SUMO Web Slides and his AMS Lecture (Jan. 2000) at
http://math.stanford.edu/~rubin/.

J. B. Tunnell, “A classical Diophantine problem and modular forms of weight $3⁄2$”, Inventiones Mathematicae, v. 72 (1983), pp. 323–334.

Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd Ed., Springer-Verlag, GTM Series.

Rational right triangles.

Similarity classes all represented by rational points on unit circle.

When a scaling is performed, area is multiplied by the square of the scaling factor.

Area of a similarity class: a positive rational modulo rational squares.

Similarity classes all represented by triangles with square-free area.

Congruent numbers: square-free representatives for the areas of similarity classes of rational right triangles.

Which square-free numbers are congruent numbers?

• A question with a long history (see Dickson's History).

• Fermat: $1$ is not a congruent number. (This question leads to the Diophantine equation $x^4+y^4=z^2$, which has no solution with positive integers.)

Parameterize via stereographic projection: $$v⁄u\to \left(u^2-v^2,2uv,u^2+v^2\right)\text{.}$$

Questions:

Which square-free numbers are congruent?

How many similarity classes for a given congruent number?

Enumerate rational numbers $v⁄u$

• Some numbers are found to be congruent:

  Enumeration Table

  v/u a b c d d (mod 8) $1⁄2$ $3$ $4$ $5$ $6$ $6$ $1⁄3$ $4$ $3$ $5$ $6$ $6$ $2⁄3$ $5$ $12$ $13$ $30$ $6$ $1⁄4$ $15⁄2$ $4$ $17⁄2$ $15$ $7$ $3⁄4$ $7⁄2$ $12$ $25⁄2$ $21$ $5$ $1⁄5$ $12$ $5$ $13$ $30$ $6$ $2⁄5$ $21$ $20$ $29$ $210$ $2$ $3⁄5$ $4$ $15⁄2$ $17⁄2$ $15$ $7$ $4⁄5$ $3⁄2$ $20⁄3$ $41⁄6$ $5$ $5$ $1⁄6$ $35$ $12$ $37$ $210$ $2$ $5⁄6$ $11$ $60$ $61$ $330$ $2$ $1⁄7$ $12$ $7⁄2$ $25⁄2$ $21$ $5$ $2⁄7$ $15$ $28⁄3$ $53⁄3$ $70$ $6$ $3⁄7$ $20$ $21$ $29$ $210$ $2$ $4⁄7$ $33⁄2$ $28$ $65⁄2$ $231$ $7$ $5⁄7$ $12$ $35$ $37$ $210$ $2$ $6⁄7$ $13$ $84$ $85$ $546$ $2$

• Congruent numbers arising with rational numbers having denominator up to 100

  There are $3043$ positive rational numbers smaller than $1$ with denominator at most $100$. Of these $1906$ give rise to the first occurrence of a (square-free) congruent number via the rational parameterization. The largest congruent number so generated from these rationals is the number $34009170$.

  d At Most 100 Occurring For Denominator Up To 100

  d   d mod (8) u/v How often? $5$ $5$ $4⁄5$ $2$ $6$ $6$ $1⁄2$ $4$ $7$ $7$ $9⁄16$ $2$ $14$ $6$ $1⁄8$ $2$ $15$ $7$ $1⁄4$ $2$ $21$ $5$ $3⁄4$ $2$ $22$ $6$ $49⁄50$ $2$ $30$ $6$ $2⁄3$ $2$ $34$ $2$ $8⁄9$ $6$ $39$ $7$ $12⁄13$ $2$ $41$ $1$ $16⁄25$ $2$ $46$ $6$ $49⁄72$ $1$ $65$ $1$ $4⁄9$ $4$ $70$ $6$ $2⁄7$ $2$ $78$ $6$ $1⁄26$ $2$ $85$ $5$ $36⁄85$ $1$

Limitations of this method

• No algorithm for finding (in finite time) if a given square-free number is congruent.

• No way to decide if a given square-free number is the congruent number for infinitely many similarity classes.

$157$ is a congruent number that arises in the enumeration based on the rational parameterization with a denominator having about 25 digits (Zagier).

Basic Proposition (Attributable to ?) If $\left(a,b,c\right)$ are postive real numbers for which $c^2=a^2+b^2$, then the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ given by $$\left\{\begin{array}{c}{x}_1=-a\left(c-a\right)⁄2\hfill \\ y_1=\pm a^2\left(c-a\right)⁄2\hfill \end{array}\right.\left\{\begin{array}{c}{x}_2=a\left(c+a\right)⁄2\hfill \\ y_2=\pm a^2\left(c+a\right)⁄2\hfill \end{array}\right.$$ lie on the plane curve $E_d$ defined by the equation $$y^2=x^3-d^{2}x\text{.}$$ Conversely, if $\left(x,y\right)$ is a point on $E_d$ with $y\ne 0$, then the triple $\left(a,b,c\right)$ given by $$a=\left|\frac{x^2-d^2}{y}\right|b=\left|\frac{2xd}{y}\right|c=\left|\frac{x^2+d^2}{y}\right|$$ is the triple of sides of a right triangle with area $d$.

If $a$, $b$, and $c$ are rational, then $E_d$ is an elliptic curve defined over the rational field. Moreover there is a 4:1 correspondence between rational points on the curve $E_d$ and rational right triangles of area $d$.

Definition. An elliptic curve defined over $\mathbf{Q}$ is the plane curve $E$ given by an equation $$y^2=x^3+ax+b$$ where $a$ and $b$ are rational and $$\Delta =-16\left(4a^3+27b^2\right)\ne 0\text{.}$$ (Within the given isomorphism class over $\mathbf{Q}$ one may assume that $a$ and $b$ are integers.)

Group law. The set $E\left(\mathbf{Q}\right)$, augmented by a single “point at infinity”, forms an abelian group in which the point at infinity is the origin and the sum of any three points lying on a line is $0$.

Theorem (Mordell, 1922). $E\left(\mathbf{Q}\right)$ is a finitely-generated abelian group.

Nagell-Lutz (1930's). If $a$ and $b$ are integers, then a point $\left(x,y\right)$ on $E$ can be a point of finite order only if $x$ and $y$ are integers and either $y=0$ or $y^2$ divides $\Delta$. .

Mazur (1977). The torsion subgroup of $E\left(\mathbf{Q}\right)$ is one of $15$ specific groups.

Corollary. The only points of finite order on the curve $E_d$ aside from the origin (at infinity) are the three points of order $2$: $\left(0,0\right)$ and $\left(\pm d,0\right)$.

Corollary. Rational right triangles of area $d$ correspond 1:4 to elements of infinite order in $E_d\left(\mathbf{Q}\right)$.

Corollary. If $d$ is a congruent number, then there are infinitely many non-similar rational right triangles with area $d$.

Question. How does one find the rank of $E\left(\mathbf{Q}\right)$ ?

Fermat. The only rational points on $E_1$ are the origin and the three points of order $2$. . Hence, the rank of $E_1\left(\mathbf{Q}\right)$ is $0$.

Core idea for studying the rank (B + S-D). For each prime $p$ let $N_p=\left|E\left({\mathbf{F}}_p\right)\right|$. Study the growth of $$\prod _{p\le x}\frac{N_p}{p}$$ as $x$ grows.

Definition. Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Its L-function $L_E\left(s\right)=L\left(E,s\right)$ is defined by $$L\left(E,s\right)=\prod _{\Delta =0modp}\frac{1}{1-c_p{p}^{-s}}\prod _{\Delta \ne 0modp}\frac{1}{1-c_p{p}^{-s}+p^{1-2s}}$$ where $$c_p=p+1-\left|E\left({\mathbf{F}}_p\right)\right|$$ when $E$ has non-singular reduction mod $p$ and otherwise, in reference to the unique singular point of the reduction mod $p$, $$c_p=\left\{\begin{array}{cc}\hfill 1& \text{ if }E\text{ has distinct rational tangents.}\hfill \\ \hfill -1& \text{ if }E\text{ has distinct irrational tangents.}\hfill \\ \hfill 0& \text{ otherwise.}\hfill \end{array}\right.$$

Theorem. The L-function of every elliptic curve defined over the rationals is the Mellin transform of a modular form of weight $2$. (Proof finished in 1999 — Shimura, Taniyama, Wiles, Breuil, Conrad, Diamond, Taylor).

Corollary. The L-function has an analytic continuation to the plane, and $L\left(E,1\right)$ “is” the product of a non-zero constant with $$\prod _{p\le x}\frac{p}{N_p}\text{.}$$

Conjecture (Birch and Swinnerton-Dyer, ca. 1960). For any elliptic curve defined over the rational field $\mathbf{Q}$ $$\mathrm{rank}E\left(\mathbf{Q}\right)={\mathrm{ord}}_{s=1}L\left(E,s\right)\text{.}$$ (The full statement is more precise.)

Fact. Not a great deal is actually known about the rank of $E\left(\mathbf{Q}\right)$ in general. Perhaps $23$ is the largest rank that is known to occur. Rubin's January AMS lecture dealt with this.

Example. $E_5\left(\mathbf{Q}\right)$ has rank $1$ with generator $\left(-4,6\right)$.

Example. $E_{34}\left(\mathbf{Q}\right)$ has rank $2$.

Theorem (Kolyvagin, 1980's)?. If $E_d\left(\mathbf{Q}\right)$ has positive rank, then $L\left(E,1\right)=0$.

For CM elliptic curves this had been shown in the 1970's by Coates and Wiles. (The curve $E_d$ is a CM curve.) In the CM case the modular curve theorem had been established by Shimura by 1970, so the analytic continuation of the L-function was known.

Theorem (Gross & Zagier, mid-1980's?). If $L\left(E,s\right)$ has a simple zero, i.e., vanishes to the first order, at $s=1$, then the rank of $E\left(\mathbf{Q}\right)$ is positive.

The work of J. B. Tunnell. Tunnell used results of Waldspurger's study of “Shimura liftings” (weight $3⁄2$) of the weight $2$ modular form corresponding to the L-function, a then-known case of the modular curve theorem established by Shimura, to compute $L\left(E_d,1\right)$.

Define $m$ and $n$ by

where $e$ is $1$ or $2$ and $e\equiv dmod2$.

Theorem (Tunnell, 1983), as formulated by Rubin. $$L\left(E_d,1\right)=\frac{{\left(m-2n\right)}^{2}e\Omega }{16\sqrt{d}}$$ where $$\Omega ={\int }_1^{\infty }\frac{dx}{\sqrt{x^3-x}}\text{.}$$

Corollary. If $d$ is congruent, then $m=2n$.

Example. When $d=1$, then $m=n=2$, and $L\left(E_1,1\right)=\frac{\Omega }{4}$. So, as Fermat showed using a different argument, $1$ is not a congruent number.

Consequence of the B + S-D Conjecture. If $m=2n$, then $d$ is congruent.

Example. If $d\equiv 5mod8$, then $m=n=0$.

If B + S-D is true, then there is an algorithm for deciding whether or not a given square-free integer $d$ is a congruent number.