2025/08/29 by Saeed Tafazolia, Tafazolia, Saeed, Jaap Top +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2509.00273
openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies hyperelliptic curves \cHd corresponding to y2=φd(x) over finite fields, with φd(x) a Chebyshev polynomial. Starting from the case where d=ℓ is an odd prime number, new cases (d,q) are presented where \cHd is maximal over the finite field \FFq2 of cardinality q2. In addition, new conditions ruling out the possibility that \cHd/\FFq2 is maximal for given (d,q), are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of \cHd with complex multiplication, and a technique from the theory of 2-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics p≡ 1\bmod 4 for d≡ 1\bmod 4 a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs (q,d) resulting in maximality.