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

Paul Monsky

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arXiv · 0908.0971

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

Paul Monsky

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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.

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Paul Monsky. 2009-08-07. Transcendence of some Hilbert-Kunz multiplicities (modulo a conjecture). https://arxiv.org/abs/0908.0971

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