Let q=p^n be a power of an odd prime, let m=(q+1)/2, and let F_i=overlineF_q^2(x,y) be the function field defined by y^m=x^i(x^2+1), with gcd(i,m)=gcd(i+2,m)=1. In the exceptional case q>7 and i=(m-2)/2, the automorphism group is conjectured to have order 4(q+1). Automorphism-group conjecture. |operatornameAut( F_i)|=4(q+1). The preceding theorem determines the automorphism group except in this exceptional case; the conjecture asserts that the automorphisms constructed there exhaust the full automorphism group. Its status is not resolved in the supplied source.

The automorphism-group conjecture for the exceptional function fields

The conjecture was formulated as Conjecture 4.10 by Peter Beelen, Maria Montanucci, Jonathan Niemann, and Luciane Quoos in 2024. In the exceptional case $q>7$ and $i=(m-2)/2$ , it asserts $|\operatorname{Aut}(\mathcal F_i)|=4(q+1)$ .

Known results

Outside the exceptional case, the automorphism group is determined.
In the exceptional case, the constructed automorphisms give a subgroup of order $4(q+1)$ , so $|\operatorname{Aut}(\mathcal F_i)|\ge 4(q+1)$ .
The unresolved step was the matching upper bound.

September 2, 2026 claimed proof

On September 2, 2026, Xu Zhuang submitted An upper bound for an exceptional automorphism group. Its abstract claims the reverse inequality and hence exact order $4(q+1)$ in the exceptional case. This is a preprint claim and has not been independently verified; no counterexample, withdrawal, or published confirmation was found.

Current status (as of September 2026): The lower bound is established, while Xu Zhuang claims the matching upper bound; the conjecture remains unverified.

The automorphism-group conjecture for the exceptional function fields

Primary source

Progress summary

Solutions 0