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

Main article: 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 $y^{2}=x^{3}-n^{2}x$, 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 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.