Descending rational points on elliptic curves to smaller fields (Q2715678)

For a given elliptic curve \(E\) over a number field \(K\) and a given finite extension \(F/K\), the authors first establish existence of a unique intermediate field \(M\) of \(F/K\) with rank\((E(M))\) equal to rank\((E(F))\). The field \(M\) is called the Minimal Subfield. The rank of \(E(F)\) is by definition the rank of the finitely generated free abelian group \(E(F)/E(F)_{tors}\) (Mordell!). If \(F/K\) is \(G\)-Galois, then \(M\) is the fixed field of the kernel of the natural group homomorphism \(G \to \Aut(E(F)/E(F)_{tors})\), so one may say that \(M\) is gotten ``by adjoining the (coordinates of) the torsion-free part of \(E(F)\) to \(K\)''. NEWLINENEWLINENEWLINEThe main purpose of the paper under review is to determine \(M\) in some cases where the rank \(r\) of \(E(F)\) is low, the general drift being that \(M/K\) is then of fairly small degree. Proposition 3 states that \([M:K]\) is bounded by a function of \(r\) alone. For \(r=1,2,3\) quite precise information concerning the possible Galois groups \(\text{Gal}(M/K)\) is provided; the main ingredient is the classification of finite subgroups of GL\((r,{\mathbb Z})\). Analogous results are obtained for curves \(E\) with CM by a maximal order \(\mathcal O\) in an imaginary quadratic field \(\mathcal K\), this time under the assumption that the rank of \(E(F)\) as an \(\mathcal O\)-module is 1 or 2. NEWLINENEWLINENEWLINEAs the authors explain, one strong motivation for looking at this problem are potential results of Coates-Wiles-Kolyvagin type: if \(F\) is abelian over \(\mathcal K\) (CM case) or \(\mathbb Q\) (modular case), then positivity of the rank of \(E(F)\) implies vanishing of \(L(E/F,s)\) at \(s=1\) by results of Arthaud, Ribet, and Kato; and if \(M\) (the Minimal Subfield) is abelian over \(\mathcal K\) (resp. \(\mathbb Q\)) then one may invoke these results, replacing \(F\) by \(M\) and maintaining the rank. (In the statement of the Coates-Wiles result on p.450 (top) one should add that \(K\) is taken to be \(\mathcal K\) or \(\mathbb Q\), so \(E\) should be defined over \(\mathcal K\).) NEWLINENEWLINENEWLINEThe last section of the paper discusses so-called elliptic analogs of Stark's theorem; the issue is, given a zero \(s=\omega\) of \(L(E/F,s)\), to find a very small subextension \(M/K\) such that \(L(E/M)\) still vanishes at \(s=\omega\). NEWLINENEWLINENEWLINEThe reviewer has one concluding remark, for whatever it is worth: The classification of minimal subfields in this paper is reminiscent of the classification of algebraic tori in low dimension [cf. early work of \textit{V. E. Voskresenskij}, Izv. Akad. Nauk SSSR, Ser. Mat. 29, 239-244 (1965; Zbl 0139.14601)].