Number of 4-cycles of the genus 2 superspecial isogeny graph - Sworski, Vladimir P., author; Pries, Rachel, advisor; Hulpke, Alexander, committee member; Rajopadhye, Sanjay, committee member; Shoemaker, Mark, committee member (Colorado State University. Libraries, 2024)
The genus 2 superspecial degree-2 isogeny graph over a finite field of size p2 is a network graph whose vertices are constructed from genus 2 superspecial curves and whose edges are the degree 2 isogenies between them. Flynn and Ti discovered 4-cycles in the graph, which pose problems for applications in cryptography. Florit and Smith constructed an atlas which describes what the neighborhood of each vertex looks like. We wrote a program in SageMath that can calculate neighborhoods of these graphs for small primes. Much of our work is motivated by these computations. We examine the prevalence of 4-cycles in the graph and, motivated by work of Arpin, et al. in the genus 1 situation, in the subgraph called the spine. We calculate the number of 4-cycles that pass through vertices of 12 of the 14 kinds possible. This also resulted in constructing the neighborhood of all vertices two steps or fewer away for three special types of curves. We also establish conjectures about the number of vertices and cycles in small neighborhoods of the spine.
Keywords: Abelian varieties; spine; 4-cycles; superspecial; isogeny graph