Tunnell's theorem

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

$\begin{matrix} A_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 2x^2 + y^2 + 32z^2 \} \\ B_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 2x^2 + y^2 + 8z^2 \} \quad \\ C_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 8x^2 + 2y^2 + 64z^2 \} \\ D_n & = & \#\{ (x,y,z) \in \mathbb{Z}^3 | n = 8x^2 + 2y^2 + 16z^2 \}. \end{matrix}$

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 $y^2 = x^3 - n^2x$, 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 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 $-\sqrt{n},\ldots,\sqrt{n}$.

References

• Koblitz, Neal (1984). Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, no. 97, Springer-Verlag. ISBN 0-387-97966-2.
• 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.