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A remark on the Mordell-Weil rank of elliptic curves over the maximal abelian extension of the rational number field. (English) Zbl 1213.11119
Summary: In this paper, we study the Mordell-Weil ranks of elliptic curves defined over the maximal abelian extension of the rational number field, assuming several conjectures on the Hasse-Weil \(L\)-functions. We prove that an elliptic curve defined over an abelian field with odd degree has infinite rank over the maximal abelian extension of the rational number field. This result gives affirmative evidence for ‘the largeness’ (in the sense of Florian Pop [Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)]) of the maximal abelian extension of the rational number field.

MSC:
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11R20 Other abelian and metabelian extensions
12E30 Field arithmetic
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References:
[1] F. Pop, Embedding problems over large fields, Ann. of Math. 144 (1996), 1-34. · Zbl 0862.12003
[2] K. A. Ribet, Torsion points of abelian varieties in cyclotomic extensions, Enseign. Math. 27 (1981), 315-319.
[3] M. Rosen and S. Wong, The Rank of Abelian Varieties over Infinite Galois Extensions, Journal of Number Theory 92 (2002), 182-196. · Zbl 1001.11025
[4] D. E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), 311-349. · Zbl 0860.11033
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