Let E/Q be an elliptic curve defined over Q of conductor N and let Gal((Q) over bar /Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism sigma epsilon. Gal((Q) over bar /Q), we let (Q) over bar sigma s be the fixed sub field of Q under s. We prove that for every s. Gal((Q) over bar /Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in (Q) over bar sigma has in finite rank, so the rank of E((Q) over bar sigma) is in finite. Our approach uses the modularity of E/Q and a collection of algebraic points on E - the so-called Heegner points - arising from the theory of complex multiplication. In particular, we show that for some integer r and for a prime p prime to rN, the rank of E over all the ring class fields of a conductor of the form rp(n) is unbounded, as n goes to infinity

- Publisher
- AMER MATHEMATICAL SOC

- Issue Date
- 2007

- Language
- English

- Article Type
- Article

- Keywords
RANK

- Citation
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.359, no.12, pp.6143 - 6154

- ISSN
- 0002-9947

- Appears in Collection
- MA-Journal Papers(저널논문)

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