Quadratic Twists of Heronian Elliptic Curves with Arbitrarily Large 2-Selmer Rank

Abstract

This paper explicitly determines the 2-Selmer group of the family of elliptic curves arising from Heron triangles of area 2mn. By analyzing the quadratic twists induced by powers of 2(varying m), we show precisely how the 2-Selmer group transforms under such twisting. For any squarefree odd integer n, the 2-Selmer rank is determined by the parity of m, the congruence class of q = (n2 + 1)/2 (mod 8), and the prime factorizations of the factors n1, n2 of n = n1n2. These results yield explicit formulas for the 2-Selmer rank and provide an explicit upper bound on the Mordell-Weil rank. In particular, this generalizes the family studied in [1] to arbitrary m, constructing explicit families of Heronian elliptic curves with prescribed and arbitrarily large 2-Selmer rank.