Rational Right Triangles and Elliptic Curves

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 $3\u20442$”,

*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: $$v\u2044u\to \left({u}^{2}-{v}^{2},\phantom{\rule{0.6em}{0ex}}2uv,\phantom{\rule{0.6em}{0ex}}{u}^{2}+{v}^{2}\right)\phantom{\rule{0.6em}{0ex}}\text{.}$$

Questions:

Enumerate rational numbers $v\u2044u$

Some numbers are found to be congruent:

**Enumeration Table****v/u****a****b****c****d****d (mod 8)**$1\u20442$ $3$ $4$ $5$ $6$ $6$ $1\u20443$ $4$ $3$ $5$ $6$ $6$ $2\u20443$ $5$ $12$ $13$ $30$ $6$ $1\u20444$ $15\u20442$ $4$ $17\u20442$ $15$ $7$ $3\u20444$ $7\u20442$ $12$ $25\u20442$ $21$ $5$ $1\u20445$ $12$ $5$ $13$ $30$ $6$ $2\u20445$ $21$ $20$ $29$ $210$ $2$ $3\u20445$ $4$ $15\u20442$ $17\u20442$ $15$ $7$ $4\u20445$ $3\u20442$ $20\u20443$ $41\u20446$ $5$ $5$ $1\u20446$ $35$ $12$ $37$ $210$ $2$ $5\u20446$ $11$ $60$ $61$ $330$ $2$ $1\u20447$ $12$ $7\u20442$ $25\u20442$ $21$ $5$ $2\u20447$ $15$ $28\u20443$ $53\u20443$ $70$ $6$ $3\u20447$ $20$ $21$ $29$ $210$ $2$ $4\u20447$ $33\u20442$ $28$ $65\u20442$ $231$ $7$ $5\u20447$ $12$ $35$ $37$ $210$ $2$ $6\u20447$ $13$ $84$ $85$ $546$ $2$ 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?**$5$ $5$ $4\u20445$ $2$ $6$ $6$ $1\u20442$ $4$ $7$ $7$ $9\u204416$ $2$ $14$ $6$ $1\u20448$ $2$ $15$ $7$ $1\u20444$ $2$ $21$ $5$ $3\u20444$ $2$ $22$ $6$ $49\u204450$ $2$ $30$ $6$ $2\u20443$ $2$ $34$ $2$ $8\u20449$ $6$ $39$ $7$ $12\u204413$ $2$ $41$ $1$ $16\u204425$ $2$ $46$ $6$ $49\u204472$ $1$ $65$ $1$ $4\u20449$ $4$ $70$ $6$ $2\u20447$ $2$ $78$ $6$ $1\u204426$ $2$ $85$ $5$ $36\u204485$ $1$

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 $$\left\{\begin{array}{c}{x}_{1}=-a\left(c-a\right)\u20442\hfill \\ {y}_{1}=\pm {a}^{2}\left(c-a\right)\u20442\hfill \end{array}\right.\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\phantom{\rule{1.8em}{0ex}}\left\{\begin{array}{c}{x}_{2}=a\left(c+a\right)\u20442\hfill \\ {y}_{2}=\pm {a}^{2}\left(c+a\right)\u20442\hfill \end{array}\right.$$ 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 $$a=\left|\frac{{x}^{2}-{d}^{2}}{y}\right|\phantom{\rule{0.3em}{0ex}}\text{}b=\left|\frac{2xd}{y}\right|\phantom{\rule{0.3em}{0ex}}\text{}c=\left|\frac{{x}^{2}+{d}^{2}}{y}\right|$$ 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 $$\Delta =-16\left(4{a}^{3}+27{b}^{2}\right)\ne 0\phantom{\rule{0.6em}{0ex}}\text{.}$$ (Within the given isomorphism class over $\mathbf{Q}$ one may assume that $a$ and $b$ are integers.)**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 $$L\left(E,s\right)\phantom{\rule{0.6em}{0ex}}=\phantom{\rule{0.6em}{0ex}}\prod _{\Delta =0modp}\frac{1}{1-{c}_{p}{p}^{-s}}\phantom{\rule{0.6em}{0ex}}\prod _{\Delta \ne 0modp}\frac{1}{1-{c}_{p}{p}^{-s}+{p}^{1-2s}}$$ 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 $p$, $${c}_{p}=\left\{\begin{array}{cc}\hfill 1& \text{if}E\text{has distinct rational tangents.}\hfill \\ \hfill -1& \text{if}E\text{has distinct irrational tangents.}\hfill \\ \hfill 0& \text{otherwise.}\hfill \end{array}\right.$$**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 $$\prod _{p\le x}\frac{p}{{N}_{p}}\phantom{\rule{0.6em}{0ex}}\text{.}$$**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 $2$.**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 $3\u20442$) 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 ${x}^{2}+2e{y}^{2}+8{z}^{2}=d\u2044e$ $n$ $=$ # integer $\left(x,y,z\right)$ with ${x}^{2}+2e{y}^{2}+32{z}^{2}=d\u2044e$ 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 $$\Omega ={\int}_{1}^{\infty}\phantom{\rule{0.6em}{0ex}}\frac{dx}{\sqrt{{x}^{3}-x}}\phantom{\rule{0.6em}{0ex}}\text{.}$$**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.