Errata for “ An Extension of Kedlaya ’ s Algorithm to Hyperelliptic Curves

In this note we correct a gap in the proof of the complexity estimates appearing in our papers [1],[2]. The complexity estimates are correct, but the proof was incomplete at a certain point. The same gap appears in our paper [3], but there the estimate for the space complexity has to be multiplied with a power of log(g), where g is the genus of the curve. First we fill in the gap in [2]. In this paper we gave an algorithm to compute the zeta function of an hyperelliptic curve of genus g over a field with q = 2 n elements. We proved that the worst-case time complexity is O(g 5+ n 3+), and that the worst-case space complexity is O(g 4 n 3). For the average-case time and space complexity we obtained O(g 4+ n 3+) and O(g 3 n 3), respectively. These estimates are correct, but there is a gap in the proof, which we will now correct. The gap in [1] can be treated in the same way. The gap has to do with the precision estimates. The matrix M of the small Frobenius (induced by squaring) on the given basis of the Monsky-Washnitzer cohomology H 1 does not have integral entries. We proved in [2] that the denominators have valuation O(log(g)). Hence the denominators in the norm M F of M have valuation at most O(n log(g)). This can cause a loss of precision of O(n log(g)) q-adic digits, as we noted in section 4.2 of [2]. What we overlooked is that in the calculation of the characteristic polynomial of M F , using the Hessenberg algorithm, there can be an accumulation of loss of precision, so that we lose g times more digits. Thus we lose at most O(ng log(g)) digits, instead of at most O(n log(g)) digits. In our paper we worked with a precision of O(ng) digits. One way to remedy this is to simply carry more precision in our calculations, from the start on. It is easy to see that this does not change the time complexity, but it would increase the space complexity by multiplying it with a power of log(g). However it is not necessary to increase our estimate for the space complexity. To prove this, we will use the following claim. Let C be a smooth proper curve over the ring Z q of q-adic integers, and let …