How do generic properties spread?

Abstract: Given a family of algebraic varieties, a natural question to ask is what type of properties of the generic fiber, and how those properties extend to other fibers. Let's explore this topic from an arithmetic point of view by looking at the scenario: Suppose we have a 1-dimensional family of pairs of elliptic curves over a number field $K$,  with the generic fiber of this family being a pair of non-isogenous elliptic curves. Furthermore, suppose the (projective) height of the parametrizer is less than or equal to $B$. One may ask how does the property of "being isogenous" extends to the special fibers. Can we give a quantitative estimation for the number of specializations of height at most $B$, such that the two elliptic curves at the specializations are isogenous?

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar