O'Nan moonshine and arithmetic

On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

“…In recent new developments of moonshine, new connections to the arithmetic of elliptic curves have been discovered [4,26,27]. In [27] for instance, the existence of a representation for the sporadic O'Nan group has been established, which controls ranks and p-torsion in Selmer groups or Tate-Shafarevich groups of quadratic twists of certain elliptic curves (for a precise statement, see [27,Theorems 1.3 and 1.4]. See also [23,Theorem 7.1].).…”

Snowmass White Paper: Moonshine

“…The phenomenon of O'Nan moonshine has been observed to have connections to the arithmetic of elliptic curves [64] as has moonshine for the Thompson group [122]. We are just beginning to scratch the surface of this intriguing connection between sporadic groups, automorphic forms, and arithmetic geometry.…”

“…The main conjecture of Mathieu and umbral moonshine was proven in [87,57] but as we explain below, there is still a great deal of mystery surrounding the origins of Mathieu and umbral moonshine. More recently there have been additional new types of moonshine: for the O'Nan sporadic group [58] and penumbral moonshine [59] which encompasses moonshine for the Thompson group [104] as a special case, much as umbral moonshine generalizes Mathieu moonshine. Although the connections of these new types of moonshine to physics are at the moment unknown, they do deserve to be called moonshine in that they are all special and finite in number and this again arises through connections to genus zero subgroups of SL(2, R) [32,40,59].…”

Hecke System of Harmonic Maass Functions and Applications to Modular Curves of Higher Genera

“…For a detailed account on moonshine, we refer the reader to a survey on classical moonshine [16] and a recent survey including umbral moonshine [13]. See also [14,15] for non-Monstrous moonshine.…”