Brandeis University Everytopic Seminar

Friday December 1, 2006, 1:40 PM, Goldsmith 226

Speaker: Paul Monsky (Brandeis)

Title: Hilbert-Kunz theory for a power series in s + 1 variables, particularly when s = 2

Abstract: Let A be a power series ring in x0,...,xs over a field k of characteristic p, f be a non-zero element of A, and q = pn. The question of the dependence on n of the colength en of the ideal generated by the (xi)q and f is subtle when s > 1. My talk will mostly deal with the case s = 2, f a form defining an irreducible plane curve. Brenner and Trivedi show that

en = μ(p2n) – Rn, with μ rational and Rn = O(pn),
and give a sheaf theoretic description of μ. I've shown that Rn = (rational-valued periodic) · pn + O(1). I'll also say something about the cases s = 2, f arbitrary, where something analogous might hold, and s > 2 which is very different indeed. I'll give lots of nice examples, some coming from the "p-fractal" theory developed by Teixeira and me.