# Outline

### References

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

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

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

### Congruent numbers

1. Rational right triangles.

2. Similarity classes all represented by rational points on unit circle.

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

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

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

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

7. 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.)

### The rational parameterization

1. Parameterize via stereographic projection: $v⁄u\to \left({u}^{2}-{v}^{2},\phantom{\rule{0.6em}{0ex}}2uv,\phantom{\rule{0.6em}{0ex}}{u}^{2}+{v}^{2}\right)\phantom{\rule{0.6em}{0ex}}\text{.}$

2. Questions:

3. 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? $5$ $5$ $4⁄5$ $2$ $6$ $6$ $1⁄2$ $4$ $7$ $7$ $9⁄16$ $2$ $14$ $6$ $1⁄8$ $2$ $15$ $7$ $1⁄4$ $2$ $21$ $5$ $3⁄4$ $2$ $22$ $6$ $49⁄50$ $2$ $30$ $6$ $2⁄3$ $2$ $34$ $2$ $8⁄9$ $6$ $39$ $7$ $12⁄13$ $2$ $41$ $1$ $16⁄25$ $2$ $46$ $6$ $49⁄72$ $1$ $65$ $1$ $4⁄9$ $4$ $70$ $6$ $2⁄7$ $2$ $78$ $6$ $1⁄26$ $2$ $85$ $5$ $36⁄85$ $1$

4. 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.

5. $157$ is a congruent number that arises in the enumeration based on the rational parameterization with a denominator having about 25 digits (Zagier).

### Connection with Elliptic Curves

1. 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}=±{a}^{2}\left(c-a\right)⁄2\hfill \end{array}\right\\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\left\{\begin{array}{c}{x}_{2}=a\left(c+a\right)⁄2\hfill \\ {y}_{2}=±{a}^{2}\left(c+a\right)⁄2\hfill \end{array}\right\$ lie on the plane curve ${E}_{d}$ defined by the equation ${y}^{2}={x}^{3}-{d}^{2}x\phantom{\rule{0.6em}{0ex}}\text{.}$ 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 is the triple of sides of a right triangle with area $d$.

2. 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 $d$.

3. 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 $\Delta =-16\left(4{a}^{3}+27{b}^{2}\right)\ne 0\phantom{\rule{0.6em}{0ex}}\text{.}$ (Within the given isomorphism class over $\mathbf{Q}$ one may assume that $a$ and $b$ are integers.)

4. 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 $0$.

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

6. 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 $\Delta$. .

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

8. 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 $\left(±d,0\right)$.

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

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

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

12. 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 $0$.

### The L-function

1. 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.

2. 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 $L\left(E,s\right)\phantom{\rule{0.6em}{0ex}}=\phantom{\rule{0.6em}{0ex}}\prod _{\Delta =0modp}\frac{1}{1-{c}_{p}{p}^{-s}}\phantom{\rule{0.6em}{0ex}}\prod _{\Delta \ne 0modp}\frac{1}{1-{c}_{p}{p}^{-s}+{p}^{1-2s}}$ 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 $p$,

3. 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).

4. 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 $\prod _{p\le x}\frac{p}{{N}_{p}}\phantom{\rule{0.6em}{0ex}}\text{.}$

5. Conjecture (Birch and Swinnerton-Dyer, ca. 1960). For any elliptic curve defined over the rational field $\mathbf{Q}$ $\mathrm{rank}\phantom{\rule{0.6em}{0ex}}E\left(\mathbf{Q}\right)\phantom{\rule{0.6em}{0ex}}=\phantom{\rule{0.6em}{0ex}}{\mathrm{ord}}_{s=1}\phantom{\rule{0.3em}{0ex}}L\left(E,s\right)\phantom{\rule{0.6em}{0ex}}\text{.}$ (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 $2$.

6. 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.

7. 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.

8. 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

 $m$ $=$ # integer $\left(x,y,z\right)$ with ${x}^{2}+2e{y}^{2}+8{z}^{2}=d⁄e$ $n$ $=$ # integer $\left(x,y,z\right)$ with ${x}^{2}+2e{y}^{2}+32{z}^{2}=d⁄e$

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 $\Omega ={\int }_{1}^{\infty }\phantom{\rule{0.6em}{0ex}}\frac{dx}{\sqrt{{x}^{3}-x}}\phantom{\rule{0.6em}{0ex}}\text{.}$

9. 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.

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

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

12. 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.