Supersingular Elliptic Curves and Moonshine

Symmetry Integrability and Geometry-Methods and Applications

article

Victor Manuel Aricheta¹ 

[1] Department of Mathematics, Emory University

关键词: Painlev´e equations;    open problems;   

DOI  :  10.3842/SIGMA.2019.007

来源: National Academy of Science of Ukraine

【 摘 要 】

We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.

【 授权许可】

Unknown