On a Galois extension with restricted ramification related to the Selmer group of an elliptic curve with complex multiplication.
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...
Some results and problems that arise in connection with the foundations of the theory of ruled and rational field extensions are discussed.