Regulators of the fixed elliptic curve over rank-one imaginary quadratic fields

We establish a one-sided central limit theorem for the logarithms of regulators of a fixed rational non-CM elliptic curve $E$ over imaginary quadratic fields of rank one, motivated by the Gross--Zagier formula and a result of Radziwiłłand Soundararajan. We also establish the existence of many imaginary quadratic fields whose regulators grow at least as fast as the square root of the conductor of the field. When the central value of the $L$-function of $E$ is non-vanishing, these results also hold for the regulator of the quadratic twist $E^{(d)}$ over $\mathbb{Q}$. Moreover, assuming the Birch and Swinnerton-Dyer conjecture, we obtain the uniform boundedness of the Shafarevich--Tate groups and Tamagawa numbers for these rank-one quadratic twists.