Title: Torsion Subgroups of CM Elliptic Curves in Degree 2p
Abstract: A common classification problem is to identify the groups
which arise as the torsion subgroup of an elliptic curve defined over
any number field of a fixed degree. That only finitely many such groups
occur is a consequence of Merel’s Uniform Boundedness Theorem. However,
for certain families of elliptic curves—such as those with complex
multiplication (CM)—recent advances have allowed us to move beyond a
fixed-degree classification to glimpse the behavior of torsion points
over infinitely many degrees of a restricted form. In this talk, I will
discuss recent work with Holly Paige Chaos which characterizes the
groups that arise as torsion subgroups of CM elliptic curves defined
over number fields of degree 2p where p is prime. Here, a
classification in the strongest sense is tied to determining whether
there exist infinitely many Sophie Germain primes.