Maximal curves of genus 5 over finite fields

Technical Summary: Maximal Curves of Genus 5 Over Finite Fields

Problem Statement
The paper investigates the existence of maximal curves of genus 5 defined over a finite field $\mathbb{F}_q$. A curve $C$ is termed maximal if the number of its $\mathbb{F}_q$-rational points, $\#C(\mathbb{F}_q)$, attains the Hasse–Weil–Serre upper bound:
$$\#C(\mathbb{F}_q) = q + 1 + g\lfloor 2\sqrt{q} \rfloor$$
where $g=5$. The study focuses specifically on the case where the discriminant of the field, defined as $d(\mathbb{F}_q) := \lfloor 2\sqrt{q} \rfloor^2 - 4q$, equals $-19$. This condition restricts the possible values of $q$ to congruence classes $0, 1, 2 \pmod 5$. The central question is whether such maximal curves exist for these specific field parameters and, if so, to provide explicit models.

Methodology
The authors employ a combination of algebraic geometry, the theory of abelian varieties, and representation theory. The methodology proceeds through the following steps:

1. Reduction to Automorphism Groups:
   Using the Honda-Tate theorem and Serre's equivalence of categories between ordinary principally polarized abelian varieties and unimodular irreducible hermitian modules, the authors establish that the Jacobian of a maximal curve of genus 5 with $d(\mathbb{F}_q) = -19$ is isogenous to the fifth power of a maximal elliptic curve $E$. Consequently, the automorphism group of the curve, $\text{Aut}_{\mathbb{F}_q}(C)$, is isomorphic to the Dihedral group $D_5$.

2. Classification of Curve Types:
   Following the Enrique-Babbage classification, a non-hyperelliptic curve of genus 5 is either trigonal (admitting a degree 3 map to $\mathbb{P}^1$) or a complete intersection of three quadrics in $\mathbb{P}^4$. The paper rules out the hyperelliptic case based on prior work by Zaytsev. The investigation then splits into analyzing these two remaining geometric structures under the constraint of the $D_5$ action.

3. Case-by-Case Analysis:

   • Case $q \equiv 2, 3, 4 \pmod 5$: The authors analyze the embedding of $D_5$ into the projective linear groups $PGL_3(\mathbb{F}_q)$ (for trigonal curves) and $PGL_5(\mathbb{F}_q)$ (for complete intersections). They demonstrate that for $q \equiv 2 \pmod 5$, the order of the group $PGL_3(\mathbb{F}_q)$ is not divisible by 5, preventing the existence of the required automorphism. For the complete intersection case, they show that the invariance of the defining quadrics under the $D_5$ action forces the intersection to be singular, contradicting the smoothness requirement of a curve.
   • Case $q \equiv 1 \pmod 5$: Here, $\mathbb{F}_q$ contains a primitive 5th root of unity. The authors utilize the theory of projective representations and central extensions to lift the $D_5$ action to linear representations. They derive potential equations for both trigonal curves (as singular plane quintics) and complete intersections of quadrics. While explicit models are constructed, computational searches using MAGMA for specific values of $q$ (e.g., 61, 311) reveal that these potential models do not yield maximal curves.
   • Case $q \equiv 0 \pmod 5$: The authors address the Diophantine equation $x^2 + 19 = 4 \cdot 5^n$ derived from the condition $d(\mathbb{F}_q) = -19$. Using $p$-adic analysis (specifically Strassmann's theorem applied to a recurrence relation), they prove that the only solution satisfying the floor function constraint is $q = 5^7 = 78125$ (Note: The text states $q=57$ in the abstract and Theorem 1.2, but the proof in Section 6 derives $q=5^7$. However, the Appendix by René Schoof, referenced in the text, concludes that for $q=57$, the curve does not exist. Correction based strictly on the provided text: The text in Section 6, Proposition 6.4, explicitly concludes "The only value of q satisfying... is q = 57." The proof derives $n=7$ leading to $5^7$, but the text asserts $q=57$. There is a discrepancy in the provided text between the derivation ($5^7$) and the stated result ($57$). However, the text explicitly states in the Abstract and Theorem 1.2 that the only value is $q=57$. The Appendix by Schoof proves non-existence for this specific case. The summary must reflect the paper's explicit claims).

Key Contributions and Results
The paper provides a comprehensive classification of the existence of maximal genus 5 curves for the discriminant $-19$:

• Non-Existence for $q \equiv 2, 3, 4 \pmod 5$: Theorem 1.1 proves that no maximal curve of genus 5 exists over $\mathbb{F}_q$ when $q \equiv 2, 3, 4 \pmod 5$ and $d(\mathbb{F}_q) = -19$. The proof relies on the incompatibility of the required automorphism group $D_5$ with the structure of the projective linear groups over these fields.
• Potential Models for $q \equiv 1 \pmod 5$: For $q \equiv 1 \pmod 5$, the authors provide explicit potential equations for both trigonal curves and complete intersections of three quadrics. These models are derived by reducing the parameters of the defining polynomials using the action of the $D_5$ automorphism group. Computational evidence suggests these candidates do not yield maximal curves for the tested fields.
• Restriction for $q \equiv 0 \pmod 5$: Theorem 1.2 establishes that $q = 57$ is the unique value satisfying $q \equiv 0 \pmod 5$ and $d(\mathbb{F}_q) = -19$. The paper notes that an appendix by René Schoof proves that even for this specific case, such a curve does not exist.

Significance
The paper contributes to the broader understanding of the "Manypoints" problem, which seeks to determine the maximum number of rational points on curves of a given genus over finite fields. By resolving the existence question for genus 5 with discriminant $-19$, the authors close a specific gap in the literature regarding small genus maximal curves. The work demonstrates the utility of combining the classification of abelian varieties (via Serre's equivalence) with geometric constraints (trigonal vs. complete intersection) and representation theory to rule out the existence of maximal curves in specific arithmetic settings. The results refine the known bounds and existence criteria for algebraic geometry codes, which rely on curves with a high number of rational points.