Elliptic Curves And Hilbert’s Tenth Problem Over Rings Of Algebraic Integers

This is a review paper.We will introduce the related results of Hilbert’s Tenth Problem over rings of algebraic integers obtained by Bjorn Poonen,B.Mazur and K.Rubin.Bjorn Poonen reduced the undecidability of Hilbert’s Tenth Problem over rings of algebraic integers to the existence of elliptic curves satisfying some properties in his paper published in 2002.B.Mazur and K.Rubin investigated the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields in their paper published in 2010.In particular,they proved that assuming the Shafarevich-Tate conjecture,the elliptic curve required by Bjorn Poonen exists,and Hilbert’s Tenth Problem over rings of algebraic integers is undecidable.