Abstract
Let E be an elliptic curve over a number field k. By the Mordell-Weil theorem the group E(K) of K-rational points on E, where K/k is a finite extension of k,is a finitely generated abelian group. We fix E/k once and for all, and we study the behavior of the rank of the group E(K) as K varies through a certain family. We are particularly interested in the family F k (G) of all Galois extensions K/k whose Galois group Gal(K/k) is isomorphic to a prescribed finite group G. In this article we focus on the case \( G = \mathbb{Z}/4\mathbb{Z} \)
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References
J. Bertin, Réseaux de Kummer et surfaces de K3, Invent. Math. 93 (1988), 267–284.
J. Fearnley, Vanishing and non-vanishing of L-series of elliptic curves twisted by Dirichlet characters, Ph.D. thesis, Concordia University, 2001.
T. Katsura, Generalized Kammer surfaces and their unirationality in characteristic p, J. Fac. Sci., Univ. Tokyo, Sect. IA 34 (1987), 1–41.
M. Kuwata, Points defined over cyclic cubic extensions on an elliptic curve and generalized Kummer surfaces, preprint.
D. E. Rohrlich, The vanishing of certain Rankin-Selberg convolutions, in “Automorphic forms and analytic number theory (Montreal, PQ, 1989),” Univ. Montréal, Montreal, QC, 123–133, 1990.
J. H. Silverman, Heights and the specialization map for families of abelian varieties,J. Reine Angew. Math. 342 (1983), 197–251.
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© 2004 Kluwer Academic Publishers
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Kuwata, M. (2004). Points Defined over Cyclic Quartic Extensions on an Elliptic Curve and Generalized Kummer Surfaces. In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_4
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DOI: https://doi.org/10.1007/978-1-4613-0249-0_4
Publisher Name: Springer, Boston, MA
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