Quadratic twists of tiling number elliptic curves

Research output: Working paper/PreprintPreprint

Authors

  • Keqin Feng
  • Qiuyue Liu
  • Jinzhao Pan
  • Ye Tian
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Details

Original languageEnglish
Publication statusE-pub ahead of print - 18 May 2024

Abstract

A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes.

Keywords

    math.NT, 11G05 (Primary) 11G40 (Secondary)

Cite this

Quadratic twists of tiling number elliptic curves. / Feng, Keqin; Liu, Qiuyue; Pan, Jinzhao et al.
2024.

Research output: Working paper/PreprintPreprint

Feng, K, Liu, Q, Pan, J & Tian, Y 2024 'Quadratic twists of tiling number elliptic curves'.
Feng, K., Liu, Q., Pan, J., & Tian, Y. (2024). Quadratic twists of tiling number elliptic curves. Advance online publication.
Feng K, Liu Q, Pan J, Tian Y. Quadratic twists of tiling number elliptic curves. 2024 May 18. Epub 2024 May 18.
Feng, Keqin ; Liu, Qiuyue ; Pan, Jinzhao et al. / Quadratic twists of tiling number elliptic curves. 2024.
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abstract = " A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes. ",
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TY - UNPB

T1 - Quadratic twists of tiling number elliptic curves

AU - Feng, Keqin

AU - Liu, Qiuyue

AU - Pan, Jinzhao

AU - Tian, Ye

N1 - 25 pages

PY - 2024/5/18

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N2 - A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes.

AB - A positive integer $n$ is called a tiling number if the equilateral triangle can be dissected into $nk^2$ congruent triangles for some integer $k$. An integer $n>3$ is tiling number if and only if at least one of the elliptic curves $E^{(\pm n)}:\pm ny^2=x(x-1)(x+3)$ has positive Mordell-Weil rank. Let $A$ denote one of the two curves. In this paper, using Waldspurger formula and an induction method, for $n\equiv 3,7\mod 24$ positive square-free, as well as some other residue classes, we express the parity of analytic Sha of $A$ in terms of the genus number $g(m):=\#2\mathrm{Cl}(\mathbb{Q}(\sqrt{-m}))$ as $m$ runs over factors of $n$. Together with $2$-descent method which express $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ in terms of the corank of a matrix of $\mathbb{F}_2$-coefficients, we show that for $n\equiv 3,7\mod 24$ positive square-free, the analytic Sha of $A$ being odd is equivalent to that $\mathrm{Sel}_2(A/\mathbb{Q})/A[2]$ being trivial, as predicted by the BSD conjecture. We also show that, among the residue classes $3$, resp. $7\mod 24$, the subset of $n$ such that both of $E^{(n)}$ and $E^{(-n)}$ have analytic Sha odd is of limit density $0.288\cdots$ and $0.144\cdots$, respectively, in particular, they are non-tiling numbers. This exhibits two new phenomena on tiling number elliptic curves: firstly, the limit density is different from the general phenomenon on elliptic curves predicted by Bhargava-Kane-Lenstra-Poonen-Rains; secondly, the joint distribution has different behavior among different residue classes.

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KW - 11G05 (Primary) 11G40 (Secondary)

M3 - Preprint

BT - Quadratic twists of tiling number elliptic curves

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