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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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