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From Picard groups of hyperelliptic curves to class groups of quadratic fields. (English) Zbl 1479.11111

Let \(C\) be a smooth projective geometrically connected curve defined over \(\mathbb{Q}\). Consider \(L\in \mathrm{Pic}^0(C)\) a line bundle, fix a point \(P\in C(\overline{\mathbb{Q}})\), and denote by \(\mathbb{Q}(P)\) the field of definition of \(P\) and \(\mathcal{O}_P\) its ring of integers. The map \(L\mapsto P^*L\) is a group morphism with respect to the group law on Pic\(^0(C)\). One wishes to specialize \(L\) into a line bundle on Spec\((\mathcal{O}_P)\) (i.e., into an ideal class of \(\mathbb{Q}(P)\)), following a question of A. Agboola and G. Pappas [Math. Res. Lett. 7, No. 5–6, 709–717 (2000; Zbl 1029.14008)]. For doing this there are two natural approaches: first by inverting primes of bad reduction of \(C\), see [J. Gillibert and A. Levin, Math. Res. Lett. 19, No. 6, 1171–1184 (2012; Zbl 1325.14019)], or secondly restrict to line bundles \(L\) corresponding to points with everywhere good reduction in the Jacobian variety \(J\) of \(C\), this second approach is the one that follows the paper under review.
Consider \(C(\mathbb{Q})\neq\emptyset\) thus \(\mathrm{Pic}^0(C)\cong J(\mathbb{Q})\). In the section two of the paper, the author defines \(P^*L\in \mathrm{Spec}(\mathcal{O}_P)\) with \(C\) an hyperelliptic curve with a rational Weierstrass point, (i.e. \(y^2=f(x)\) over \(\mathbb{Q}\) with \(\deg(f)\) odd). The author uses any regular flat model \(\mathcal{X}\) of \(C\) over \(\mathrm{Spec}(\mathbb{Z})\), and by use of Poincaré line bundle \(\mathcal{P}_{\mathcal{X}}\) on \(\mathcal{X}\times \mathrm{Pic}^0_{\mathcal{X}/\mathbb{Z}}\), define the specialization \(P^*L_Q\in \mathrm{Pic}(\mathcal{O}_P)\) by pullback of \(\mathcal{P}_{\mathcal{X}}\) at \((P,Q)\) where \(Q\in \mathcal{J}^0(\mathbb{Z})\subset J(\mathbb{Q})\) the connected component of the Néron model associated to rigidified line bundles on \(\mathcal{X}\) (which degree zero in the components in this setting), and \(L_Q\in \mathrm{Pic}^0(C)\) the corresponding line bundle of \(Q\). The author, in the paper under review, proves that \[P^*L_Q=\langle P,Q\rangle^{cl}\] where the pairing is the Mazur-Tate class group pairing [B. Mazur and J. Tate, Prog. Math. 35, 195–237 (1983; Zbl 0574.14036)].
Now, is well known that elements in the class group of quadratic imaginary extensions of \(\mathbb{Q}\) interplay with binary quadratic forms. Thus, for hyperelliptic curves, the author puts attention with points \((n,\sqrt{f(n)})\) belonging to an imaginary quadratic field, we recall \(\deg(f)\) is odd. M. M. Wood [Adv. Math. 226, No. 2, 1756–1771 (2011; Zbl 1262.11049)] proved that invertible modules over double cover of schemes can be described by binary forms and the author makes it explicit for the affine curve \(R[y]/(y^2-D)\) with \(R\) a ring such that every locally free \(R\)-modules of finite rank is free. Such description for \(\mathrm{Pic}(C\setminus\infty)\) is given by D. G. Cantor [Math. Comput. 48, 95–101 (1987; Zbl 0613.14022)] or D. Mumford [Tata lectures on theta. II: Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura. Basel: Birkhäuser (2007; Zbl 1112.14003)]. They gave an explicit quadratic form over \(\mathbb{Q}[x]\) with discriminant \(4f(x)\) associated to the hyperelliptic curve \(C:y^2=f(x)\). The author provides an integral version for elements of \(\mathrm{Pic}(\mathcal{W})\) as binary quadratic forms over \(\mathbb{Z}[X]\), where \(\mathcal{W}\) is an integral affine scheme over \(\mathrm{Spec}(\mathbb{Z})\), defined by \(y^2=f(x)\). After such preparation the author is able to control such binary quadratic forms under specialization to obtain no trivial elements of the class group in the corresponding imaginary quadratic field, more precisely:
1.
for each \(n\in\mathbb{Z}\) such that \(f(n)<0\), \(T_n:\mathrm{Spec}(\mathbb{Z}[\sqrt{f(n)}])\rightarrow \mathcal{W}\) (the closed subscheme of \(\mathcal{W}\) defined by \(x=n\)), and any \(\mathcal{L}\in \mathrm{Pic}(\mathcal{W})\) (with generic fiber \(L\in \mathrm{Pic}(C\setminus\infty)\), related with a quadratic form) non-trivial, exist infinitely many \(n\in\mathbb{Z}_{<0}\) such that \(T_n^*\mathcal{L}\neq 0\), in particular if \(\mathcal{L}\) has infinite order then the order of \(T_n^*\mathcal{L}\) is unbounded when \(n\rightarrow-\infty\).
2.
let \(P_n\in C(\overline{\mathbb{Q}})\) be the generic fiber of the section \(T_n\), assume \(f\in\mathbb{Z}[x]\) is a square-free monic polynomial whose irreducible factors have degree at most three or the abc conjecture holds, then given \(Q\in \mathcal{J}^0(\mathbb{Z})\) of infinite order, the order of \(\langle P_n,Q\rangle^{c}\) is unbounded when \(n\rightarrow-\infty\).

We recall for elliptic curves there are similar results obtained by R. Soleng [J. Number Theory 46, No. 2, 214–229 (1994; Zbl 0811.14035)], thus the paper extends such results to hyperelliptic curves under some hypothesis on the Weierstrass points or that the abc conjecture holds. Also relates Soleng’s class group morphism and Mazur-Tate’s class groups pairing, extending results of D. A. Buell and G. S. Call [J. Number Theory 167, 31–73 (2016; Zbl 1416.11076)] to hyperelliptic curves. Thus last chapter of the paper under review provide a relation between the work of the paper with the work previously done for hyperelliptic curves [T. K. Sivertsen and R. Soleng, J. Number Theory 131, No. 12, 2303–2309 (2011; Zbl 1268.11087)], and with the case of elliptic curves and class groups by D. A. Buell [Math. Comput. 30, 610–623 (1976; Zbl 0334.12003); J. Lond. Math. Soc., II. Ser. 15, 19–25 (1977; Zbl 0355.12006)] and Soleng [ Zbl 0811.14035].

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11E12 Quadratic forms over global rings and fields
14H40 Jacobians, Prym varieties
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References:

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