The Monsky matrices and non-congruent numbers (Q6546514)

Let \(n\) be a fixed positive square-free integer and let \(E_i\) and \(E_i'(i = 1, 2,3)\) be the elliptic curves defined by\N\begin{align*}\N&E_1: y^2=x^3-n^2x,\quad &E_1': y^2=x^3+4n^2x,\\\N&E_2: y^2=x(x+n)(x+2n),\quad &E_2': y^2=x^3-6nx^2+n^2x,\\\N&E_3: y^2=x(x-n)(x-2n),\quad &E_3': y^2=x^3+6nx^2+n^2x.\N\end{align*}\NFor simplicity, we also use the symbol \(E\) to denote the congruent elliptic curve \(E_1\). The congruent number problem is to decide if a positive integer is the area of a rational right triangle. It is well known that \(n\) is a non-congruent number if and only if the congruent elliptic curve \(E\) has the Mordell-Weil rank zero.\N\NLet \(S^{(2)}(E/\mathbb{Q})\) denote the 2-Selmer group of the congruent elliptic curve \(E\). Define \(s(n) = \log_2(|S^{(2)}(E/\mathbb{Q})|)-2\). Then \(s(n)\) is called the 2-Selmer rank of \(E\). Let \(r(n)\) be the Mordell-Weil rank of \(E\) over \(\mathbb{Q}\). Let \(\Sha(E/\mathbb{Q})\) be the Shafarevich-Tate group of \(E\), and \(t(n)\) the 2-rank of the group \(\Sha(E/\mathbb{Q})[2]\). \(n\) is a non-congruent number if \(s(n)=0\) or \(s(n)=t(n)(> 0)\).\N\NLagrange presented a lot of non-congruent numbers \(n\) with at most three odd prime factors by using\Nthe 2-descent method. Following Lagrange's method, Feng, Xiong and Xue obtained some equivalent conditions for \(E_i,E_i' (i = 1,2,3)\) to have Selmer groups with the lowest orders simultaneously and consequently found a series of non-congruent numbers with arbitrarily many prime divisors. It turns out that \(s(n) = 0\) for these\Nnon-congruent numbers. Many numbers were constructed such that \(s(n) = 0\), and hence these numbers are non-congruent numbers. \textit{D. Li} and \textit{Y. Tian} [Acta Math. Sin., Engl. Ser. 16, No. 2, 229--236 (2000; Zbl 1080.11500)] obtained a sufficient condition for \(n\) to be a non-congruent number whose prime factors are all congruent to 1 modulo 8 such that \(E\) has 2-Selmer group with order 16. \textit{Y. Ouyang} and \textit{S. Zhang} [Sci. China, Math. 57, No. 3, 649--658 (2014; Zbl 1364.11108)] extended the result of Li and Tian to those \(n\) whose prime factors are all congruent to 1 modulo 4. \textit{H. Qin} [Math. Ann. 383, No. 3--4, 1647--1686 (2022; Zbl 1503.11141)] developed a method which can be used to deal with the case where the 2-Sylow subgroup of\N\(\Sha(E)\) is \(\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}\). It is worth noting that in these cases \(s(n) = t(n) = 2\). \textit{Z. Wang} [Sci. China, Math. 59, No. 11, 2145--2166 (2016; Zbl 1410.11056)] gave a sufficient condition such that \(\Sha(E/\mathbb{Q})[2] \cong(\mathbb{Z}/2\mathbb{Z})^{s(n)}\) for \(n\) whose prime factors are all congruent to 1 modulo 8 and \(s(n)\geq 4\).\N\NIn this paper, the authors explicitly calculated the Selmer groups for elliptic curves \(E_i(i = 1,2,3)\), and obtained the results on the Selmer groups in the papers of \N\textit{F. Lemmermeyer} [Acta Arith. 110, No. 1, 15--36 (2003; Zbl 1047.11055)] and \textit{F. Lemmermeyer} and \textit{R. Mollin} [Acta Math. Univ. Comen., New Ser. 72, No. 1, 73--80 (2003; Zbl 1097.11012)]. They also obtained many equivalent conditions for \(s(n) = 0\) such that both \(E_i\) and \(E_i'\) have Selmer groups with the lowest orders simultaneously, together with some criteria for \(n\) with arbitrarily large \(s(n)\) and whose odd prime factors may be congruent to 3 modulo 4 to be a non-congruent number. The results can cover\Nmany criteria as special cases. The approach is to combine the computation of the rank of Monsky matrix and the method developed in the papers of Li, Tian and Ouyang, Zhang.\N\NThere are many results in this paper. Due to the notations are too many to display, we just give the following corollary.\N\NCorollary 1.2. Let \(n = p_1 \cdots p_kq_1\cdots q_t \equiv 1 \pmod {8}, n_1 = q_1\cdots q_t \equiv 1 \pmod {8},k+t = m\), where \(p_i \equiv 1 \pmod {4}, 1 \leq i \leq k\), and \(q_j \equiv 3 \pmod {4}, 1 \leq j \leq t\). Assume that \(\big(\frac{q_j}{p_i}\big)= 1\) for \(1 \leq i \leq k, 1 \leq j \leq t\), and at least one of \(q_j \equiv 3 \pmod {8}\). Let \(\mathrm{rank}(A_{k\times k})=-s(n)\). If there exist \(2^{s(n)}-3\) elements \(d\) in \(U = \{d(\vec{v}):A_{k\times k}\vec{v}=\vec{0}\}\) such that \(\delta(d, n)=1\), then \(\mathrm{rank}_{\mathbb{Z}}(E/\mathbb{Q}) = 0\) and \(\Sha(E/\mathbb{Q})[2] \cong(\mathbb{Z}/2\mathbb{Z})^{s(n)}\). In particular, \(n\) is a non-congruent number.