2024/05/18 by Keqin Feng, Feng, Keqin, Qiuyue Liu +5 · 1 citation
Computer Science · Mathematics · #11G05 (Primary) 11G40 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2405.11132
openalex publication_date 2024/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A positive integer n is called a tiling number if the equilateral triangle can be dissected into nk2 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(± n):± ny2=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≡ 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):=#2Cl(ℚ(√(-m))) as m runs over factors of n. Together with 2-descent method which express dim_\mathbbF2Sel2(A/ℚ)/A[2] in terms of the corank of a matrix of \mathbbF2-coefficients, we show that for n≡ 3,7\mod 24 positive square-free, the analytic Sha of A being odd is equivalent to that Sel2(A/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⋯ and 0.144⋯, 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.