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Bounds of the rank of the Mordell-Weil group of Jacobians of hyperelliptic curves. (English. French summary) Zbl 1475.11120

Let \(C/\mathbb{Q}\) be a hyperelliptic curve given by a model \(y^2 = f(x)\), with \(f(x) \in \mathbb{Q}[x]\), and let \(J/\mathbb{Q}\) be its Jacobian. The Mordell-Weil theorem states that \(J(\mathbb{Q})\) is a finitely generated abelian group and hence \(J(\mathbb{Q})\) decomposes as a direct sum \(J(\mathbb{Q})_{\text{tors}} \oplus \mathbb{Z}^{R_{J(\mathbb{Q})}}\), where \(J(\mathbb{Q})_{\text{tors}}\) is the subgroup of torsion elements and \(R_{J(\mathbb{Q})}\) is the rank of \(J(\mathbb{Q})\). During the last decades a great amount of research has gone into finding bounds of \(R_{J(\mathbb{Q})}\) in terms of invariants of \(C\). In this article the authors give families of examples of hyperelliptic curves \(C \colon y^2 = f(x)\) defined over \(\mathbb{Q}\), with \(f(x)\) of degree \(p\), where \(p\) is a Sophie Germain prime, such that \(R_{J(\mathbb{Q})}\) is bounded by the genus of \(C\) and the two-rank of the class group of the cyclic field defined by \(f(x)\). They further exhibit examples where the given bound is sharp. This extends work of D. Shanks [Math. Comput. 28, 1137–1152 (1974; Zbl 0307.12005)] and L. C. Washington [Math. Comput. 48, 371–384 (1987; Zbl 0613.12002)] where a similar bound is given for the rank of certain elliptic curves.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
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References:

[1] Armitage, John V.; Fröhlich, Albrecht, Classnumbers and unit signatures, Mathematika, 14, 94-98 (1967) · Zbl 0149.29501 · doi:10.1112/S0025579300008044
[2] Cassels, J. W. S., Arithmetic and geometry, Vol. I, 35, The Mordell-Weil group of curves of genus \(2, 27-60 (1983)\), Birkhäuser · Zbl 0529.14015
[3] Daniels, Harris B.; Lozano-Robledo, Álvaro; Wallace, Erik, Data for Bounds of the rank of the Mordell-Weil group of Jacobians of Hyperelliptic Curves
[4] Davis, Daniel, Computing the number of totally positive circular units which are squares, J. Number Theory, 10, 1, 1-9 (1978) · Zbl 0369.12002 · doi:10.1016/0022-314x(78)90002-1
[5] Dummit, David S.; Voight, John, The \(2\)-Selmer group of a number field and heuristics for narrow class groups and signature ranks of units (2017) · Zbl 1457.11152
[6] Edgar, Hugh M.; Mollin, Richard A.; Peterson, Brian L., Class groups, totally positive units, and squares, Proc. Am. Math. Soc., 98, 1, 33-37 (1986) · Zbl 0604.12007 · doi:10.2307/2045762
[7] Estes, Dennis R., On the parity of the class number of the field of \(q\) th roots of unity, Rocky Mt. J. Math., 19, 3, 675-682 (1989) · Zbl 0703.11052 · doi:10.1216/rmj-1989-19-3-675
[8] Garbanati, Dennis A., Unit signatures, and even class numbers, and relative class numbers, J. Reine Angew. Math., 274/275, 376-384 (1975) · Zbl 0312.12015 · doi:10.1515/crll.1975.274-275.376
[9] Gras, Marie-Nicole, Non monogénéité de l’anneau des entiers des extensions cycliques de \({\mathbb{Q}}\) de degré premier \(\ell \ge 5\), J. Number Theory, 23, 3, 347-353 (1986) · Zbl 0564.12008 · doi:10.1016/0022-314X(86)90079-X
[10] Hughes, I.; Mollin, Richard A., Totally positive units and squares, Proc. Am. Math. Soc., 87, 4, 613-616 (1983) · Zbl 0509.12004 · doi:10.2307/2043345
[11] Kim, Myung-Hwan; Lim, Sung-Geun, Square classes of totally positive units, J. Number Theory, 125, 1, 1-6 (2007) · Zbl 1148.11055 · doi:10.1016/j.jnt.2006.04.010
[12] Lozano-Robledo, Álvaro, A probabilistic model for the distribution of ranks of elliptic curves over \(\mathbb{Q} (2016)\) · Zbl 1405.11076
[13] Marcus, Daniel A., Number fields, viii+279 p. pp. (1977), Springer · Zbl 0383.12001
[14] Oriat, Bernard, Relation entre les \(2\)-groupes des classes d’idéaux au sens ordinaire et restreint de certains corps de nombres, Bull. Soc. Math. Fr., 104, 3, 301-307 (1976) · Zbl 0352.12007
[15] Poonen, Bjorn; Schaefer, Edward F., Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math., 488, 141-188 (1997) · Zbl 0888.11023
[16] Schaefer, Edward F., \(2\)-descent on the Jacobians of hyperelliptic curves, J. Number Theory, 51, 2, 219-232 (1995) · Zbl 0832.14016 · doi:10.1006/jnth.1995.1044
[17] Shanks, Daniel, The simplest cubic fields, Math. Comput., 28, 1137-1152 (1974) · Zbl 0307.12005 · doi:10.2307/2005372
[18] Stevenhagen, Peter, Class number parity for the \(p\) th cyclotomic field, Math. Comput., 63, 208, 773-784 (1994) · Zbl 0819.11050 · doi:10.2307/2153298
[19] Stoll, Michael, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith., 98, 3, 245-277 (2001) · Zbl 0972.11058 · doi:10.4064/aa98-3-4
[20] Washington, Lawrence C., Class numbers of the simplest cubic fields, Math. Comput., 48, 177, 371-384 (1987) · Zbl 0613.12002 · doi:10.2307/2007897
[21] Washington, Lawrence C., Introduction to cyclotomic fields, 83, xiv+487 p. pp. (1997), Springer · Zbl 0966.11047 · doi:10.1007/978-1-4612-1934-7
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