JOURNAL OF
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COMBINATORICS

Home > Vol 23, No 2 (2006)

Fu Liu¹ and Brian Osserman²

¹Massachusetts Institute of Technology Massachusetts Massachusetts
²University of California Berkeley Berkeley

Mochizuki's work on torally indigenous bundles [1] yields combinatorial identities by degenerating to different curves of the same genus. We rephrase these identities in combinatorial language and strengthen them, giving relations between Ehrhart quasi-polynomials of different polytopes. We then apply the theory of Ehrhart quasi-polynomials to conclude that the number of dormant torally indigenous bundles on a general curve of a given type is expressed as a polynomial in the characteristic of the base field. In particular, we conclude the same for the number vector bundles of rank two and trivial determinant whose Frobenius-pullbacks are maximally unstable, as well as self-maps of the projective line with prescribed ramification.

1. S. Mochizuki, Foundations of p-adic Teichm\ddot uller theory, American Mathematical Society, 1999.
2. B. Osserman, Mochizuki's crys-stable bundles: A lexicon and applications, to appear in Publications of the Research Institute for Mathematical Sciences.
3. B. Osserman, Rational functions with given ramification in characteristic p, to appear in Compositio Mathematica.
4. R. P. Stanley, Enumerative Combinatorics: Vol. I, Cambridge Studies in Advanced Mathematics, no. 49, Cambridge University Press, 1997.