A Remark on the Mordell-Weil Rank of Elliptic Curves over the Maximal Abelian Extension of the Rational Number Field

Abstract

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 Pop) of the maximal abelian extension of the rational number field.