The Iwasawa theory of the elliptic curve $y^2=x^3x$ Prof. John H. Coates Cambridge University, UK. 131207 Abstract The ancient Eastern problems about congruent numbers amount to questions about the arithmetic of the curve in the title over the fields $F = Q (\sqrt{D}$), where $D$ is a square free integer $>1$. We shall briefly recall these, and then discuss what noncommutative Iwasawa theory tells us about the arithmetic of $E$ over the fields $$F_n = Q (D^{\frac{1}{p^n}}) \quad (n=1,2,\cdots )$$ where $p$ is an odd prime, and $D$ is a $p$power free integer $>1$. As we shall explain at the end, the phenomena which occur in this special case appear to be part of a very general theory involving the arithmetic of motives over $p$adic Lie extensions of number fields
