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Constructing distinct curves with isomorphic Jacobians in characteristic zero. (English) Zbl 0832.14019
According to Humbert, Hayashida and Nishi, and Lange there exist nonisomorphic curves over the complex numbers whose Jacobians are isomorphic as (nonpolarized) abelian varieties. The construction of such examples consists of the construction of suitable period matrices for the corresponding Jacobians rather than the equations of the curves themselves. Now the paper under review gives examples of curves with isomorphic Jacobians by means of equations for the curves in question. The main result is:
Let \(m\) be a positive even square-free integer with \(n\) odd prime divisors, let \(h\) be the class number of the ring \(\mathbb{Z} [\sqrt {-m}]\). Let \(f \in \mathbb{Z}[x]\) be the minimal polynomial of the \(j\)-invariant of the elliptic curves with complex multiplication by \(\mathbb{Z} [\sqrt {-m}]\) and let \(S\) be the set of positive real roots of the polynomial \(g = (x + 1)^hf ({2^8 x^3 \over x + 1})\). Then the set \(\{y^2 = x^6 - kx^4 + kx^2 - 1 |k \in S\}\) consists of \(2^n\) pairwise nonisomorphic curves, all of whose Jacobians are isomorphic to one another as unpolarized abelian varieties.

MSC:
14H40 Jacobians, Prym varieties
14H42 Theta functions and curves; Schottky problem
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