Counting points on Cab curves using Monsky-Washnitzer cohomology

We describe an algorithm to compute the zeta function of any C"a"b curve over any finite field F"p"^"n. The algorithm computes a p-adic approximation of the characteristic polynomial of Frobenius by computing in the Monsky-Washnitzer cohomology of the curve and thus generalizes Kedlaya's algorithm for hyperelliptic curves. For fixed p the asymptotic running time for a C"a"b curve of genus g over F"p"^"n is O(g^5^+^@?n^3^+^@?) and the space complexity is O(g^3n^3).

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