A note on rank one quadratic twists of elliptic curves and the non-degeneracy of

We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such alttext="3"> 3 encoding="application/x-tex">3 is an Eisenstein prime of good reduction, a positive proportion the quadratic twists psi"> ψ encoding="application/x-tex">E_{\psi } E"> encoding="application/x-tex">E have Mordell–Weil rank one and -adic height pairing on psi Baseline left-parenthesis Q right-parenthesis"> ( stretchy="false">) }(\mathbb {Q}) non-degenerate. also similar but weaker results other primes. The method proof yields examples middle codimensional algebraic cycles over number fields arbitrarily large dimension (generalized Heegner cycles) non-zero alttext="p"> p encoding="application/x-tex">p height. It not known – though expected archimedian these higher-codimensional non-zero.