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Congruent numbers, quadratic forms and \(K_2\)

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Abstract

We give a new and self-contained proof for an alternative form of Tunnell’s theorem on congruent numbers. The proof relies on more knowledge on quadratic forms with less calculation than Tunnell’s proof. The approach is based on our observation that the equality \(\#\{(x,y,z,w) \in \mathbb {Z}^4|p=2x^2+3y^2+3z^2+4w^2+2yz\}=\#\{(x,y,z) \in \mathbb {Z}^3|p^2=2x^2+4y^2+9z^2-4yz\}\) holds for any odd prime p. The same method is applied to the elliptic curves \(E_n:~y^2=x^3+n^2x\) and \(E_{n}:~~y^2=x(x-n)(x+3n)(E_{-n}:~~y^2=x(x+n)(x-3n))\)(\(\pi /3\)(\(2\pi /3\))-congruent elliptic curves), where we list, without proof, the analogous results. A series of criteria for congruent numbers are given. In particular, for a prime p, we show that if \(p\equiv 1\pmod {8}\) is a congruent number then the 8-rank of \(K_2O_{\mathbb {Q}(\sqrt{p})}\) equals one;  if \(p\equiv 1\pmod {16}\) with \(h(-p)\not \equiv h(-2p)\pmod {16}\) then 2p is not a congruent number; and if \(p\equiv 1, q\equiv 3\pmod {8}\) are two primes with \(h(-pq)\not \equiv h(-p)\pmod {8}\) then pq is not a congruent number.

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Acknowledgements

The author would like to thank the referees for careful reading and many valuable comments and suggestions. In particular, he would like to thank the referees for letting him know the existence of Li and Zhao’s paper [30]. He would also like to thank Professors J. W. Hoffman and Y. Tian for their help.

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Hourong Qin

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Communicated by Wei Zhang.

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H. Qin: Supported by NSFC (Nos.11571163, 11631009), SQ2020YFA070208.

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Qin, H. Congruent numbers, quadratic forms and \(K_2\). Math. Ann. 383, 1647–1686 (2022). https://doi.org/10.1007/s00208-021-02263-x

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