On the Mordell-Weil lattice of the elliptic curve \(y^2=x^3+t^m+1\). II (Q2769543)

In Part I [ibid. 49, 71--78 (2000; Zbl 0990.11036)], the author studied the Mordell-Weil lattice \(L_m\) of the elliptic curve \(E^{(m)}:\, y^2=x^3+t^m+1\) over the field \(k(t)\), where \(k\) is an algebraically closed field of characteristic zero. Put \(A=\{9, 12, 18, 24, 30, 60\}\). The main result of Part I was that all the \(L_m\) can be described by the \(L_i\) for \(i\in A\), and some well-known root lattices. In the present article, the author studies the lattices \(L_i\) for \(i\in A\). A prime number \(p\) is called a splitting prime of \(E^{(m)}\) if \(E^{(m)}({\mathbb F}_p(t))=E^{(m)}(\overline{\mathbb F}_p(t))\). According to Shioda, if \(p\equiv 1\) mod \(6m/(m,6)\), then \(E^{(m)}(\overline{\mathbb F}_p(t))\cong L_m\). The author finds splitting primes for \(L_i\) for \(i\in A, i\neq 60\). In the cases \(m=9\) and \(m=12\), he gives a detailed description of the lattice structure of \(L_m\), including the number of minimal sections, the Gram matrix, the determinant, and the center density.