Generators of the automorphism group of the Fermat quartic surface

The problem asks for explicit generators of the full automorphism group of the Fermat quartic surface, not merely its finite subgroup preserving the standard polarization. Earlier work established the polarization-preserving group but left the full infinite group without a known generating set.

Known results

• The finite polarization-preserving subgroup was determined as $\operatorname{Aut}(X_4,H_4)=\widetilde{F}_{384}$, of order $2^9\cdot 3$ (2003).
• The full group was shown to be infinite (2003).
• An algorithmic study of the complex Fermat quartic reported that computing the full group was too heavy to complete and that its automorphism group remained unknown (2013).
• In characteristic $3$, Shimada generated the automorphism group using the polarization-preserving subgroup together with two involutions; this is a different setting from the complex problem.

August 19, 2026 claimed solution

A new arXiv preprint, “Sixteen generators of the automorphism group of the Fermat quartic surface,” claims an explicit generating set of $16$ elements for the full group. This is a claimed resolution, but the available record identifies it as an unrefereed preprint and supplies no independent verification or response to possible gaps.

Current status (as of August 2026): An explicit $16$-generator description is claimed in a new preprint, while the claim remains unverified and the problem is not yet settled.