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Ranks of twists of elliptic curves and Hilbert’s tenth problem. (English) Zbl 1227.11075
Let \(K\) be a number field, \(E/K\) an elliptic curve over \(K\), \(\text{Sel}_2(E/K)\) the \(2\)-Selmer group, \(d_2(E/K):=\dim_{{\mathbb F}_2}\text{Sel}_2(E/K)\), \[ N_r(E,X):=|\{F/K\mid [F:K]\leq 2,\, d_2(E^F/K)=r,\, N_{K/{\mathbb Q}}{\mathfrak f}(F/K)<X \}|, \] where \(E^F\) is the quadratic twist of \(E\) by \(F/K\), \({\mathfrak f}(F/K)\) the finite part of the conductor of \(F/K\). The authors conjecture that if \(r\geq \dim_2 E(K)[2]\) and \(r\equiv d_2(E/K)\pmod 2\), then \(N_r(E,X)\gg X\); if \(K\) has a real embedding, or if \(E/K\) does not acquire everywhere good reduction over an abelian extension of \(K\), then \(N_r(E,X)\gg X\) for every \(r\geq\dim_{{\mathbb F}_2}E(K)[2]\). Suppose that \(E\) has a quadratic twist \(E'/K\) with \(d_2(E'/K)=r\). The authors prove that if \(\text{Gal}(K(E[2])/K)\simeq S_3\), then \[ N_r(E,X)\gg \frac{X}{(\log X)^{2/3}}, \] and if \(\text{Gal}(K(E[2])/K)\simeq {\mathbb Z}/3{\mathbb Z}\), then \[ N_r(E,X)\gg \frac{X}{(\log X)^{1/3}}. \] Suppose that \(E(K)[2]=0\) and that either \(K\) has a real embedding, or that \(E\) has multiplicative reduction at some prime of \(K\). If \(r=0,1\), or \(r\leq d_2(E/K) \), then \(E\) has many quadratic twists \(E'/K\) with \(d_2(E'/K)=r \). Other sufficient conditions for the existence of twists of arbitrary \(2\)-Selmer rank are presented. If \(L/K\) is a cyclic extension of prime degree, then there is an elliptic curve \(E/K\) with \(\text{rank}(E(L))=\text{rank}(E(K))\). If the Shafarevich-Tate conjecture is true, then there is an elliptic curve \(E\) over \(K\) with \(\text{rank}(E(L))=\text{rank}(E(K))=1\). As applications of these results: there is an elliptic curve \(E\) over \(K\) with \(E(K)=0\); if the Shafarevich-Tate conjecture is true for every number field \(K\), then Hilbert’s Tenth Problem has a negative answer over every infinite ring that is finitely generated over \(\mathbb Z\).

MSC:
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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