The action of the Frobenius map on rank vector bundles in characteristic Y Laszlo and C Pauly

The action of the Frobenius map on rank 2 vector bundles in characteristic 2 Y. Laszlo and C. Pauly March 1, 2001 1 Introduction Let X be a smooth algebraic curve of genus g over a field k of characteristic p > 0. The behaviour of semi-stable bundles with respect to the absolute Frobenius Fa remains mysterious if g ≥ 2. Let us briefly explain why this question should be of interest. Start with a continuous representation ? of the algebraic fundamental group in GLr(k¯), where k¯ is the algebraic closure of k. Let E? be the corresponding rank r bundle over X. Then all the bundles F (n)?a E?, for n > 0, where F (n)a denotes the n-fold composite Fa ? · · · ? Fa, are semi-stable. Conversely, assuming that k is finite, let E be a semi-stable rank r bundle defined over k¯. Because the set of isomorphism classes of semi-stable bundles of degree 0 over Xk? , where k? is any finite extension of k, is finite, one observes (see [LS]) that some twist of E comes from a representation as above. Therefore, if one is interested in unramified continuous representations of the Galois group over k of a global field k(X) in characteristic p, it is natural to look at Frobenius semi-stable bundles, that is those whose pull-backs by F (n)a are all semi-stable.

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