Tunnell's theorem

Short description: On the congruent number problem: which integers are the area of a rational right triangle

In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.

Congruent number problem

The congruent number problem asks which positive integers can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions of a few fairly simple Diophantine equations.

Theorem

For a given square-free integer n, define

$\begin{array}{rl}{A}_n& =\mathrm{\#}\{(x,y,z)\in {\mathbb{\mathbb{Z}}}^3\mid n=2x^2+y^2+32z^2\},\\ B_n& =\mathrm{\#}\{(x,y,z)\in {\mathbb{\mathbb{Z}}}^3\mid n=2x^2+y^2+8z^2\},\\ C_n& =\mathrm{\#}\{(x,y,z)\in {\mathbb{\mathbb{Z}}}^3\mid n=8x^2+2y^2+64z^2\},\\ D_n& =\mathrm{\#}\{(x,y,z)\in {\mathbb{\mathbb{Z}}}^3\mid n=8x^2+2y^2+16z^2\}.\end{array}$

Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2A_n = B_n and if n is even then 2C_n = D_n. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form $y^2=x^3-n^{2}x$, these equalities are sufficient to conclude that n is a congruent number.

History

The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in (Tunnell 1983).

Importance

The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation. For instance, for a given $n$, the numbers $A_n,B_n,C_n,D_n$ can be calculated by exhaustively searching through $x,y,z$ in the range $-\sqrt{n},\dots ,\sqrt{n}$.

See also

• Birch and Swinnerton-Dyer conjecture
• Congruent number

References

• Koblitz, Neal (2012), Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics (Book 97) (2nd ed.), Springer-Verlag, ISBN 978-1-4612-6942-7 
• Tunnell, Jerrold B. (1983), "A classical Diophantine problem and modular forms of weight 3/2", Inventiones Mathematicae 72 (2): 323–334, doi:10.1007/BF01389327, http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002099403