Papers › Counting points on superelliptic curves in average polynomial time
Counting points on superelliptic curves in average polynomial time
Andrew V. Sutherland
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We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over ℚ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves yᵐ=f(x) with m≥2 and f∈ℤ[x] any squarefree polynomial of degree d≥3, along with a positive integer N. It can compute #X(𝔽ₚ) for all p≤N not dividing mlc(f)disc(f) in time O(md³ Nlog³ NloglogN). It achieves this by computing the trace of the Cartier--Manin matrix of reductions of X. We can also compute the Cartier--Manin matrix itself, which determines the p-rank of the Jacobian of X and the numerator of its zeta function modulo~p.
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