2016/03/02 by Yih-Dar Shieh, Shieh, Yih-Dar
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1603.00566
openalex publication_date 2016/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe an algorithm to compute the zeta function of any non-hyperelliptic genus 3 plane curve C over a finite field with automorphism group G = ℤ / 2 ℤ. This algorithm computes in the Monsky-Washnitzer cohomology of~the curve. Using the relation between the Monsky-Washnitzer cohomology of C and its quotient E := C/G, the computation splits into 2 parts: one in a subspace of the Monsky-Washnitzer cohomology and a second which reduces to the point counting on an elliptic curve E. The former corresponds to the dimension 2 abelian surface ker(Jac(C) → E), on which we can compute with lower precision and with matrices of smaller dimension. Hence we obtain a faster algorithm than working directly on the curve C.