From Picard groups of hyperelliptic curves to class groups of quadratic fields

Gillibert, Jean

doi:10.1090/tran/8334

From Picard groups of hyperelliptic curves to class groups of quadratic fields
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Trans. Amer. Math. Soc. 374 (2021), 3919-3946 Request permission

Abstract:

Let $C$ be a hyperelliptic curve defined over $\mathbb {Q}$, whose Weierstrass points are defined over extensions of $\mathbb {Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree $0$ on $C$ which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas.

References

A. Agboola and G. Pappas, Line bundles, rational points and ideal classes, Math. Res. Lett. 7 (2000), no. 5-6, 709–717. MR 1809295, DOI 10.4310/MRL.2000.v7.n6.a4
Manjul Bhargava, Higher composition laws. II. On cubic analogues of Gauss composition, Ann. of Math. (2) 159 (2004), no. 2, 865–886. MR 2081442, DOI 10.4007/annals.2004.159.865
Andrew R. Booker and T. D. Browning, Square-free values of reducible polynomials, Discrete Anal. , posted on (2016), Paper No. 8, 16. MR 3533307, DOI 10.19086/da.732
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Duncan A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610–623. MR 404205, DOI 10.1090/S0025-5718-1976-0404205-X
Duncan A. Buell, Elliptic curves and class groups of quadratic fields, J. London Math. Soc. (2) 15 (1977), no. 1, 19–25. MR 434949, DOI 10.1112/jlms/s2-15.1.19
Duncan A. Buell and Gregory S. Call, Class pairings and isogenies on elliptic curves, J. Number Theory 167 (2016), 31–73. MR 3504033, DOI 10.1016/j.jnt.2016.02.030
Gregory Scott Call, LOCAL HEIGHTS ON FAMILIES OF ABELIAN VARIETIES, ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–Harvard University. MR 2634990
David G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), no. 177, 95–101. MR 866101, DOI 10.1090/S0025-5718-1987-0866101-0
David A. Cox, Primes of the form $x^2 + ny^2$, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
Ofer Gabber, Qing Liu, and Dino Lorenzini, The index of an algebraic variety, Invent. Math. 192 (2013), no. 3, 567–626. MR 3049930, DOI 10.1007/s00222-012-0418-z
Jean Gillibert and Aaron Levin, Pulling back torsion line bundles to ideal classes, Math. Res. Lett. 19 (2012), no. 6, 1171–1184. MR 3091601, DOI 10.4310/MRL.2012.v19.n6.a1
Andrew Granville, $ABC$ allows us to count squarefrees, Internat. Math. Res. Notices 19 (1998), 991–1009. MR 1654759, DOI 10.1155/S1073792898000592
Alexander Grothendieck, Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim.
Christopher Hooley, On the square-free values of cubic polynomials, J. Reine Angew. Math. 229 (1968), 147–154. MR 220686, DOI 10.1515/crll.1968.229.147
Martin Kneser, Composition of binary quadratic forms, J. Number Theory 15 (1982), no. 3, 406–413. MR 680541, DOI 10.1016/0022-314X(82)90041-5
Qing Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577–4610 (French, with English summary). MR 1363944, DOI 10.1090/S0002-9947-96-01684-4
B. Mazur and J. Tate, Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195–237. MR 717595
Laurent Moret-Bailly, Groupes de Picard et problèmes de Skolem. I, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 161–179, 181–194 (French). MR 1005158
David Mumford, Tata lectures on theta. II, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura; Reprint of the 1984 original. MR 2307768, DOI 10.1007/978-0-8176-4578-6
George Pólya, Über ganzwertige ganze Funktionen, Rend. Circ. Mat. Palermo 40 (1915), no. 1, 1–16.
C. S. Seshadri, Triviality of vector bundles over the affine space $K^{2}$, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456–458. MR 102527, DOI 10.1073/pnas.44.5.456
Tormod Kalberg Sivertsen and Ragnar Soleng, Hyperelliptic curves and homomorphisms to ideal class groups of quadratic number fields, J. Number Theory 131 (2011), no. 12, 2303–2309. MR 2832825, DOI 10.1016/j.jnt.2011.05.002
Ragnar Soleng, Homomorphisms from the group of rational points on elliptic curves to class groups of quadratic number fields, J. Number Theory 46 (1994), no. 2, 214–229. MR 1269253, DOI 10.1006/jnth.1994.1013
Melanie Matchett Wood, Gauss composition over an arbitrary base, Adv. Math. 226 (2011), no. 2, 1756–1771. MR 2737799, DOI 10.1016/j.aim.2010.08.018
Melanie Matchett Wood, Parametrization of ideal classes in rings associated to binary forms, J. Reine Angew. Math. 689 (2014), 169–199. MR 3187931, DOI 10.1515/crelle-2012-0058