2025/02/19 by Peter Beelen, Beelen, Peter, Maria Montanucci +3
Computer Science · Mathematics · #14H05 #14H37 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2502.13815
openalex publication_date 2025/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we complete the work started in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known \mathbbFq2-maximal function field Z3 having the third largest genus, for q ≡ 0 \pmod 3. The cases q ≡ 2 \pmod 3 and q ≡ 1 \pmod 3 have been in fact analyzed in arXiv:2303.00376v1 [math.AG] and arXiv:2404.18808v1 [math.AG], respectively. As in the other two cases, the function field Z3 arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, Z3 has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of \mathbbFq2-rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, Aut(Z3) is exactly the automorphism group inherited from the Hermitian function field, apart from the case q=3.