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Ranks in the family of hyperelliptic Jacobians of \(y^2= x^5+ax\). II. (English) Zbl 1497.11159

Let \(a\in\mathbb{N}\) be a \(8\)-th power free and consider the hyperelliptic curve \(C_{a}:\;y^2=x^5+ax\). Moreover, let \(J_{a}\) denote the Jacobian variety connected to \(C_{a}\). In this paper, the author continues his study of the properties of \(J_{a}\). In a previous paper [PartI, the author, J. Number Theory 223, 35–52 (2021; Zbl 1468.11135)] the case of irreducible polynomial \(x^4+a\) was considered. In the case when \(x^4+a\) is a reducible polynomial in \(\mathbb{Q}[x]\), i.e., when \(a=-u^2, a=-u^4\) or \(a=4u^4\), then, under some mild conditions, an upper bound for the rank of \(J_{a}\) is obtained. For example, if \(a=-u^4\), where \(u\) is squarefree and each prime factor is \(\equiv3\pmod{4}\), then \(\operatorname{rank}J_{a}(\mathbb{Q})\leq 3\omega(2a)\), where \(\omega(n)\) counts the number of prime divisors of an integer \(n\). Similar results are obtained for \(a=-u^2, a=-4u^4\). In the case \(a=-u^2\) some additional assumptions concerning the arithmetic of the quadratic field \(\mathbb{Q}(\pm\sqrt{u})\) are needed. In some special cases the result are strong enough to compute the rank exactly. For example, in the case \(a=-u^4, u\in\{p, 2p\}\), \(p\) prime and \(p\equiv 3\pmod{4}\) we have the equality \(\operatorname{rank}J_{a}(\mathbb{Q})=0\) and this can be used to compute the set of rational points on \(C_{a}\) (in the mentioned example we have \(C_{a}(\mathbb{Q})=\{\infty, (0,0), (\pm b, 0)\}\)).

MSC:

11G10 Abelian varieties of dimension \(> 1\)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G20 Curves over finite and local fields
11G25 Varieties over finite and local fields

Citations:

Zbl 1468.11135

Software:

Magma
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References:

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