Monstrous Moonshine emerged in the 1970s, from coincidences relating the largest
sporadic simple group to moduli spaces of complex elliptic curves. More
recently, manifestations of moonshine have materialized that relate sporadic
simple groups to arithmetic invariants of elliptic curves over the rationals. In
this talk I will describe forthcoming joint work with Cheng and Mertens that
initiates a systematic approach to this phenomena. Our investigations yielded
some unexpected results, including a connection between the congruent number
problem of antiquity, and the smallest sporadic simple group.
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