Abstract
Let be a
-isogeny class of elliptic curves defined over
without CM. The isogeny graph associated to
is a graph which has a vertex for each elliptic curve in
and an edge for each
-isogeny of prime degree that maps one elliptic curve in
to another elliptic curve in
, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over
of the corresponding elliptic curve. Then, the main statement of the article is a classification of the 2-adic Galois image that occurs at each vertex of all isogeny-torsion graphs consisting of elliptic curves defined over
without CM.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
The author would like to express his utmost gratitude to Álvaro Lozano-Robledo, Jeremy Rouse, Drew Sutherland, and David Zureick-Brown for their patience and many helpful conversations on this topic. The author would also like to thank the referee for many comments that improved the paper.