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 3/2”,

*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 ----> (u^{2} - v^{2}, 2 u v, u^{2} + v^{2}) . Questions:

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 (a, b, c) are postive real numbers for which c^{2} = a^{2} + b^{2}, then the points (x_{1}, y_{1}) and (x_{2}, y_{2}) given by{ x_{1} = -a (c - a)/2 y_{1} = ± a^{2} (c - a)/2 { x_{2} = a (c + a)/2 y_{2} = ± a^{2} (c + a)/2 lie on the plane curve E_{d} defined by the equation

y^{2} = x^{3} - d^{2} x . Conversely, if (x, y) is a point on E_{d} with y ≠ 0, then the triple (a, b, c) given by

a = |{x^{2} - d^{2}}/{y}| b = |{2 x d}/{y}| c = |{x^{2} + d^{2}}/{y}| 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**Q**is the plane curve E given by an equationy^{2} = x^{3} + a x + b where a and b are rational and

**Delta**= -16 (4 a^{3} + 27 b^{2}) ≠ 0 .(Within the given isomorphism class over

**Q**one may assume that a and b are integers.)**Group law.**The set E(**Q**), 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(**Q**) is a finitely-generated abelian group.**Nagell-Lutz (1930's).**If a and b are integers, then a point (x, y) 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(**Q**) 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: (0, 0) and (± d, 0).**Corollary.**Rational right triangles of area d correspond 1:4 to elements of infinite order in E_{d}(**Q**).**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(**Q**) ?**Fermat.**The only rational points on E_{1} are the origin and the three points of order 2. . Hence, the rank of E_{1}(**Q**) is 0.

**Core idea for studying the rank (B + S-D).**For each prime p let N_{p} = |E(**F**_{p})| . Study the growth ofPROD_{p <= x}[{{N_{p}}/{p}} ] as x grows.

**Definition.**Let E be an elliptic curve defined over**Q**. Its**L-function**L_{E}(s) = L(E, s) is defined byL(E, s) = PROD_{ **Delta**= 0 mod p}[{{1}/{1 - c_{p} p^{-s}}}] PROD_{**Delta**≠ 0 mod p}[{{1}/{1 - c_{p} p^{-s} + p^{1 - 2s}}}]where

c_{p} = p + 1 - | E( **F**_{p})| when E has non-singular reduction mod p and otherwise, in reference to the unique singular point of the reduction mod p,

c_{p} = { 1 if E has distinct rational tangents. -1 if E has distinct irrational tangents. 0 otherwise. **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(E, 1) “is” the product of a non-zero constant withPROD_{p <= x}[{{p}/{N_{p}}} ] . **Conjecture (Birch and Swinnerton-Dyer, ca. 1960).**For any elliptic curve defined over the rational field**Q**rank E( **Q**) = ord_{s = 1} L(E, s) .(The full statement is more precise.)

**Fact.**Not a great deal is actually known about the rank of E(**Q**) in general. Perhaps 23 is the largest rank that is known to occur. Rubin's January AMS lecture dealt with this.**Example.**E_{5}(**Q**) has rank 1 with generator (-4, 6).**Example.**E_{34}(**Q**) has rank 2.**Theorem (Kolyvagin, 1980's)?.**If E_{d}(**Q**) has positive rank, then L(E, 1) = 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(E, s) has a simple zero, i.e., vanishes to the first order, at s = 1, then the rank of E(**Q**) 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(E_{d}, 1).Define m and n by

m = # integer (x, y, z) with x^{2} + 2 e y^{2} + 8 z^{2} = d/e n = # integer (x, y, z) with x^{2} + 2 e y^{2} + 32 z^{2} = d/e where e is 1 or 2 and e

**EQUIV**d mod 2.**Theorem (Tunnell, 1983), as formulated by Rubin.**L(E_{d}, 1) = {(m - 2 n)^{2} e **Omega**}/{16 SQRT{d}}where

**Omega**= INT_{1}^{INFTY}[ {{dx}/{SQRT{x^{3} - x}}} ] .**Corollary.**If d is congruent, then m = 2 n.**Example.**When d = 1, then m = n = 2, and L(E_{1}, 1) = {**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 = 2 n, then d is congruent.**Example.**If d**EQUIV**5 mod 8, 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.