# 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   ---->   (u^{2} - v^{2},  2 u v,  u^{2} + v^{2})    .

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 (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.

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 Q is the plane curve E given by an equation

 y^{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.)

4. 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.

5. Theorem (Mordell, 1922). E(Q) is a finitely-generated abelian group.

6. 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. .

7. Mazur (1977). The torsion subgroup of E(Q) 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: (0, 0) and (± d, 0).

9. Corollary. Rational right triangles of area d correspond 1:4 to elements of infinite order in E_{d}(Q).

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(Q) ?

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}(Q) is 0.

### The L-function

1. Core idea for studying the rank (B + S-D). For each prime p let N_{p} = |E(F_{p})| . Study the growth of

 PROD_{p  <=  x}[{{N_{p}}/{p}} ]

as x grows.

2. Definition. Let E be an elliptic curve defined over Q. Its L-function L_{E}(s) = L(E, s) is defined by

 L(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.

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(E, 1) “is” the product of a non-zero constant with

 PROD_{p  <=  x}[{{p}/{N_{p}}} ]    .

5. 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.

6. 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.

7. 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.

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(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}}} ]    .

9. 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.

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

11. Example. If d EQUIV 5 mod 8, 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.