Abelian surfaces over finite fields containing no curves of genus $3$ or less

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no curves of genus up to $2$ initiated by the first author \emph{et al.~}in previous work. Secondly, we show that, for simple abelian surfaces, containing a curve of genus $3$ is equivalent to admitting a polarisation of degree $4$. Thanks to this result, we can use existing algorithms to check which isomorphism classes in the isogeny classes containing no genus $2$ curves have a polarisation of degree $4$. Thirdly, we characterise isogeny classes of abelian surfaces with no curves of genus $\leq 2$, containing no abelian surface with a polarisation of degree $4$. Finally, we describe the absolutely irreducible genus $3$ curves lying on abelian surfaces containing no curves of genus less than or equal to $2$.

💡 Research Summary

The paper investigates abelian surfaces defined over a finite field 𝔽_q and asks which of them contain no (possibly singular) curves of genus three or less. The authors build on earlier work that classified isogeny classes of abelian surfaces lacking curves of genus ≤ 2, and they extend the classification to the genus‑3 case.

First, the authors recall that an isogeny class of an abelian surface over 𝔽_q is uniquely determined by its Weil polynomial
 f(t)=t⁴+at³+bt²+aqt+q².
They introduce two families of Weil polynomials: P₍irr₎ⁿᵖᵖ, corresponding to isogeny classes that are not principally polarizable, and P₍irr₎ʷʳᵉˢ, corresponding to Weil restrictions of elliptic curves defined over the quadratic extension 𝔽_{q²}. The reducible polynomials (t²−2)² and (t²−3)² are treated separately because they give rise to isogeny classes that contain a curve of arithmetic genus 2 but no absolutely irreducible curve of genus 2.

Main Theorem 1 gives a complete description of when an isogeny class contains no absolutely irreducible curve of geometric genus ≤ 2. The theorem is phrased in three equivalent ways: (i) geometric genus ≤ 2 curves are absent, (ii) arithmetic genus ≤ 2 curves are absent, and (iii) a mixed situation where absolutely irreducible genus ≤ 2 curves are absent but an arithmetic genus 2 curve exists over 𝔽_q. The conditions translate into explicit constraints on the coefficients a and b of f(t); for instance, a = 0 and b = 1−2q characterise the Weil‑restriction case.

The second major contribution is Main Theorem 2, which establishes a striking equivalence for simple abelian surfaces: the surface admits a polarization of degree 4 if and only if it contains an 𝔽_q‑irreducible curve of arithmetic genus 3. This result links the existence of a low‑degree polarization to the presence of a low‑genus curve and provides a practical criterion: checking for a degree‑4 polarization is enough to decide whether a genus‑3 curve can lie on the surface.

Using this equivalence, the authors exploit an algorithm (due to the third author in a previous paper) that, given an ordinary isogeny class, enumerates all isomorphism classes of abelian surfaces admitting a degree‑4 polarization. Consequently they can decide, for each isogeny class in P₍irr₎ⁿᵖᵖ ∪ P₍irr₎ʷʳᵉˢ, whether any member contains a genus‑3 curve. The algorithm works by computing the endomorphism ring, testing whether it is maximal, and then searching for suitable polarizations.

Main Theorem 3 provides a full classification of those isogeny classes that contain no surface with a degree‑4 polarization. The theorem is phrased in terms of the CM field K = ℚ