Non-isogenous elliptic curves and hyperelliptic Jacobians (Q6100654)

Suppose that \(K\) is a field of characteristic different from 2 and \(\bar{K}\) an algebraic closure of \(K\). Let \(f(x)\) and \(h(x)\) be cubic polynomials over \(K\) without repeated roots so that exactly one of them is irreducible. We denote by \(C_f\) and \(C_h\) the elliptic curves defined by \(y^2 = f(x)\) and \(y^2 = h(x)\). The aim of this paper is to discuss when \(C_f\) and \(C_h\) are not isogenous over \(\bar{K}\), using only the properties of \(f(x)\) and \(h(x)\) over \(K\). So, if \(C_f\) and \(C_h\) are isogenous over \(\bar{K}\), then it is proved that the \(\mathbb{Q}\)-algebras \(\mathrm{End}(C_f)\otimes \mathbb{Q}\) and \(\mathrm{End}(C_h)\otimes \mathbb{Q}\) contain a subfield isomorphic to \(\mathbb{Q}(\sqrt{-3})\), and in case where \(\mathrm{char}(K) = 0\) the elliptic curves \(C_f\) and \(C_h\) are isogenous over \(\bar{K}\) to the elliptic curve \(y^2 = x^3 - 1\). Furthermore, if \(f(x), h(x)\in K[x]\) are cubic polynomials without repeated roots such that their splitting fields are linearly disjoint over \(K\), \(h(x)\) is irreducible over \(K\), the splitting field of \(f(x)\) has degree 6 over \(K\), and the elliptic curves \(C_f\) and \(C_h\) are isogenous over \(\bar{K}\), then it is proved that \(p = char(K)\) is a prime that is not congruent to 1 modulo 3, and both \(C_f\) and \(C_h\) are supersingular elliptic curves. Several interesting corollaries of the above results are given. These results are deduced from more general results about nonisogenous hyperelliptic jacobians. More precisely, if \(n\) is an odd prime such that 2 is a primitive root \(\bmod \ n\), \(f(x), h(x) \in K[x]\) are degree \(n\) polynomials without repeated roots so that only one of them is irreducible, and the Jacobians of curves \(C_f : y^2 = f(x)\) and \(C_h : y^2 = h(x)\) are isogenous over \(\bar{K}\), then both Jacobians are abelian varieties of CM type over \(\bar{K}\) with multiplication by the \(n\)-th cyclotomic field \(\mathbb{Q}(\zeta_n)\). Moreover, the case when both polynomials are irreducible while their splitting fields are linearly disjoint is investigated.