A note on the growth of Mordell-Weil ranks of elliptic curves in cyclotomic \(\mathbb Z_p\)-extensions. (English) Zbl 1186.11031

Summary: In this note, we exhibit some examples of elliptic curves whose Mordell-Weil ranks grow in lower layer of the cyclotomic \(\mathbb Z_p\)-extension over the rationals.


11G05 Elliptic curves over global fields
11R23 Iwasawa theory
Full Text: DOI Euclid


[1] Chinta, G.: Analytic ranks of elliptic curves over cyclotomic fields. J. Reine Angew. Math., 544 , 13-24 (2002). · Zbl 1028.11040
[2] Greenberg, R.: Introduction to Iwasawa theory for elliptic curves. Arithmetic Algebraic Geometry (Park City, UT, 1999), IAS/Park City Math. Ser. 9, Amer. Math. Soc., Providence, pp. 407-464 (2001). · Zbl 1002.11048
[3] Matsuno, K.: Mordell-Weil group of an elliptic curve associated with a polynomial of degree five. (In preparation).
[4] Rohrlich, D. E.: Realization of some Galois representations of low degree in Mordell-Weil groups. Math. Research Letters, 4 , 123-130 (1997). · Zbl 0881.11052
[5] Shioda, T.: An infinite family of elliptic curves over \(\Q\) with large rank via Néron’s method. Invent. Math., 106 , 109-119 (1991). · Zbl 0766.14024
[6] Silverman, J. H.: Heights and the specialization map for families of abelian varieties. J. Reine Angew. Math., 342 , 197-211 (1983). · Zbl 0505.14035
[7] Ulmer, D.: Elliptic curves with large rank over function fields. Ann. of Math., 155 , 295-315 (2002). · Zbl 1109.11314
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.