# Transcendence of some Hilbert-Kunz multiplicities (modulo a conjecture)

## TL;DR

This paper explores the potential transcendence of certain Hilbert-Kunz multiplicities in characteristic 2, contingent on a conjecture, linking these multiplicities to a known transcendental number.

## Contribution

It demonstrates that assuming a specific conjecture, transcendental Hilbert-Kunz multiplicities can exist, connecting them to a known transcendental series.

## Key findings

- Assuming the conjecture, transcendental multiplicities are possible.
- A specific transcendental series is expressed as a Q-linear combination of Hilbert-Kunz multiplicities.
- The work links algebraic geometry with transcendental number theory.

## Abstract

Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2 hypersurfaces has algebraic but not necessarily rational Hilbert-Kunz multiplicities. We now show that if the conjecture holds, then transcendental multiplicities exist, and in particular that the number \sum\binom{2n}{n}^{2}/(65,536)^{n}, proved transcendental by Schneider, is a Q-linear combination of Hilbert-Kunz multiplicities of characteristic 2 hypersurfaces.