The Erdős–Selfridge Supersingularity Problem

open

Posed by attributed to multiple sources in the 1970s modular forms community · 1975 · algebraic geometry / number theory

The problem

Question: are there infinitely many primes p such that every modular form of weight 2 and level 1 vanishes mod p? Equivalently: are there infinitely many supersingular j-invariants over 𝔽_p?

References

sourcehttps://arxiv.org/abs/math/0409180

History & significance

Kaneko and Zagier conjectured the count equals ⌊p/12⌋ + ε(p) for explicit ε. Numerical evidence is overwhelming through millions of primes. The connection to monster group moonshine via supersingular j-values (Ogg noticed the primes dividing the monster order exactly) makes this a bridge between number theory and conformal field theory.

Still open.

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