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Infinite families of pairs of curves over \(\mathbb{Q}\) with isomorphic Jacobians. (English) Zbl 1093.14041
Although Torelli’s theorem states that a curve is completely determined by its polarized Jacobian variety, it is long known that unpolarized Jacobians may not reflect the whole geometry of the curve. For instance it is shown in [E. W. Howe, Proc. Am. Math. Soc. 129, 1647–1657 (2001; Zbl 0974.14021)] that there exists a pair of genus 3 curves over \(\mathbb{C}\) for which we cannot decide whether they are hyperelliptic by just looking at the corresponding Jacobian varieties. However, this example lacks the arithmetic flavor in the sense that none of the curves is actually defined over \(\mathbb{Q}\); and moreover for any given number field only finitely many examples can be defined over that number field.
In the paper under review, the author produces three new examples of non-isomorphic curves with isomorphic non polarized Jacobian varieties. Each family is parametrized by an open subset of \(\mathbb{P}^1\), so each family gives an infinite number of examples over \(\mathbb{Q}\). The first family consists of pairs of genus two curves whose equations are simple expressions in terms of the parameter, the curves of this family have reducible Jacobians. The second family also consists of pairs of genus two curves, but generically the curves of the family have absolutely simple Jacobian varieties. The third example consists of pairs of genus three curves, one member of each pair is a hyperelliptic curve, whereas the other a plane quartic. These examples show for instance that it is not possible in general to tell from a genus two curve over \(\mathbb{Q}\) whether the curve has rational points (or even real points). Using this result Q. Liu, D. Lorenzini and M. Raynaud [Invent. Math. 157, 455–518 (2004; Zbl 1060.14037)] were able to prove that Jacobians of genus 2 curves do not determine the number of components on the reduction of a minimal model of the curve modulo a prime. The families are constructed using methods which depend on previous work of the author, F. LeprĂ©vost and B. Poonen [Forum Math. 12, 315–364 (2000; Zbl 0983.11037)] as well as on P. R. Bending’s explicit description [Curves of genus two with \(\sqrt{2}\) multiplication, arXiv:math.NT/9911273] of genus two curves whose Jacobian varieties have real mutiplication by \(\mathbb{Z}[\sqrt{2}]\).

14H40 Jacobians, Prym varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H45 Special algebraic curves and curves of low genus
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