Glueing principally polarized abelian varieties in wild characteristic and explicit equations for extremal genus-3 curves over finite fields

Lauter's 2002 paper (with Serre's appendix) on the maximum and minimum number of rational points on genus-3 curves over finite fields left four open threads. Two of them are now closed by external evidence: the Diophantine exclusion $p\ne 5$ is vacuous (by the Luca-Tengely-Togbé theorem as applied by Ahrens-Grantham, the only perfect power $y^k$ with $k\ge 3$ of the form $x^2+x+3$ is $243=3^5$), and the borderline classification of Serre's category equivalence over non-maximal orders is settled by Jordan-Keeton-Poonen-Rains-Shepherd-Barron-Tate (2018). The emblematic explicit-construction instance $q=243$ is closed by Mestre's effective construction of a genus-3 curve over $\mathbb{F}_{243}$ with 334 points. The two open threads addressed in this task are: a general theory of glueing principally polarized abelian varieties along kernels whose order is divisible by the characteristic (the wild case), and explicit equations or effective algorithms for the extremal genus-3 curves whose existence is still only established abstractly, in the generality of Lauter's unconditional existence theorem for all finite fields.

Background

For a smooth projective geometrically irreducible curve $C$ of genus $g$ over the finite field $\mathbb{F}_q$, the Weil bound, sharpened by Serre, limits the number $N(C)$ of $\mathbb{F}_q$-rational points to $N(C)\le q+1+g\lfloor 2\sqrt{q}\rfloor$. Writing $m=\lfloor 2\sqrt{q}\rfloor$, the defect of $C$ is $q+1+gm-N(C)$, and $N_q(g)$ denotes the maximum of $N(C)$ over genus-$g$ curves over $\mathbb{F}_q$. Determining $N_q(g)$ for fixed small $g$ combines the arithmetic of the Frobenius endomorphism with the geometry of the moduli of curves: by Torelli's theorem the Jacobian of $C$ is a principally polarized abelian variety (ppav) of dimension $g$ carrying an indecomposable principal polarization, so upper bounds follow from classifying which isogeny classes of abelian varieties contain such Jacobians, while lower bounds require producing explicit curves.

For $g=3$ this program was pushed far by Lauter (2002), with an appendix by Serre. Genus-3 curves are either hyperelliptic or non-singular plane quartics; their Jacobians are abelian threefolds. Serre's appendix establishes, when $d=a^2-4q$ is a fundamental discriminant (the discriminant of the imaginary quadratic field $\mathbb{Q}(\sqrt{d})$ itself, so that the order $R_d=\mathbb{Z}[\pi]$ generated by the Frobenius $\pi$ of $E$ is the maximal order; in Lauter's applications $d$ is moreover of class number one, e.g. $d\in\{-3,-11\}$), an equivalence of categories between abelian varieties over $\mathbb{F}_q$ isogenous to a power of an ordinary elliptic curve $E$ (one whose Frobenius trace is prime to $q$) and torsion-free finite-type modules over $R_d$, polarizations translating into positive definite Hermitian forms; Lauter used this dictionary to glue polarizations on products of elliptic curves and to rule out, via "Serre's obstruction" (the non-existence of a suitable indecomposable Hermitian module), defect-0 and defect-2 zeta types (a zeta type is the factorization pattern of the Weil polynomial of the Jacobian into elliptic-curve factors, written $[m,m,m-2]$ when the Jacobian is isogenous to $E_m\times E_m\times E_{m-2}$, where $E_t$ denotes an elliptic curve over $\mathbb{F}_q$ with $q+1+t$ points). Her main theorems are unconditional: for $q=x^2+x+a$ with $a\in\{1,3\}$, $a\le x$, one has $N_q(3)\le q+1+3m-3$ and the bound is attained up to sign (Theorem 1); for $q=x^2+b$, $b\in\{1,2\}$, $b\le x$, the defect is at most 2 (Theorem 2); and for every $q$ there exists a genus-3 curve with $|N(C)-(q+1)|\ge 3m-3$ (Theorem 3). The paper nevertheless left four explicitly acknowledged open issues: (a) the arguments for excluding defect 2 in genus 3 excluded $p=5$ provisionally, because a hypothetical solution of $5^e=x^2+x+3$ with $e\ge 3$ odd and $5\mid(2x-1)$ (only excluded by a computer search with PARI, a number-theory computer algebra system, for $e<1600$) could invalidate them, and the exceptional field $q=343$ needed separate treatment; (b) in the exceptional field $q=243$ and in other cases where existence is proved only abstractly via Hermitian-module or glueing arguments, no explicit plane quartic equations were produced - an exhaustive search over plane quartics over $\mathbb{F}_{243}$ was estimated at about $3^{40}$ steps, and the suggested route was to write down the elliptic factors $E_m$, $E_{m-3}$ and glue them explicitly; (c) the glueing of polarizations along kernels whose order is divisible by the characteristic (the "wild" case, arising at $q=243$ where the kernels are $3$-torsion of étale-local type $\mathbb{Z}/3\mathbb{Z}\times\mu_3$, a product of the constant étale group scheme $\mathbb{Z}/3\mathbb{Z}$ with the multiplicative group scheme $\mu_3$ of cube roots of unity, so that part of the kernel is infinitesimal) was handled only case by case, with no general theory; and (d) the functor $S$ of the appendix (which turns a module back into an abelian variety) was used even when $R_d=\mathbb{Z}[\pi]$ is a non-maximal order in the ring of integers of $\mathbb{Q}(\sqrt{d})$, where the equivalence of categories was not known to hold.

Subsequent literature closed several of these gaps. Luca, Tengely and Togbé (2009) solved the Lebesgue-Nagell-type equation $x^2+C=4y^n$ for all $1\le C\le 100$ with $C\equiv 3\pmod 4$ and $\gcd(x,y)=1$, $n\ge 3$; for $C=11$ the unique solution is $31^2+11=4\cdot 3^5$. Ahrens and Grantham (2025, preprint) combined this with classical results of Lebesgue, Ljunggren, Nagell and Bugeaud-Mihăilescu to show that the only perfect power $y^k$, $k\ge 3$, of the form $x^2+x+3$ is $243=3^5$; hence no solution of $5^e=x^2+x+3$ with $e\ge 3$ exists and Lauter's $p\ne 5$ hypothesis is vacuous, so her defect-2 exclusion holds unconditionally (the field $q=343$ had already been handled inside the paper via the Waterhouse classification of possible Frobenius traces of elliptic curves). Jordan, Keeton, Poonen, Rains, Shepherd-Barron and Tate (2018) proved a necessary and sufficient classification of exactly when the Hom-functors give an equivalence of categories between modules over $R=\mathrm{End}\,E$ and abelian varieties isogenous to powers of $E$ - for ordinary $E$ precisely when $\mathbb{Z}[\pi]=\mathrm{End}\,E$ - positioning Serre's appendix as the maximal-order case and closing the borderline question (d), with algorithmic continuations by Marseglia. Mestre (2010, preprint), using plane quartics with an $S_3$-automorphism group, gave an effective construction of a genus-3 curve over $\mathbb{F}_{243}$ with $3^5+1+3m-3=334$ points, resolving the emblematic instance of (b) together with the max-versus-min sign ambiguity that Lauter had left open; the manYPoints registry (the online database of record for $N_q(g)$ values) records $N_{243}(3)=334$ and $N_{343}(3)=452$. Ritzenthaler's survey (2011) describes the wider explicit program - characteristic-2 families, Serre's analytic strategy via the Siegel modular form $\chi_{18}$ (a weight-18 modular form on the moduli of principally polarized abelian threefolds which, in characteristic $\ne 2$, is a nonzero square exactly on non-hyperelliptic Jacobians and vanishes exactly on hyperelliptic ones, so that being a square characterizes Jacobians), and universal families of explicit equations for optimal curves whose elliptic factors have complex multiplication (endomorphism ring an order in an imaginary quadratic field) by class-number-one discriminants - while noting that large endomorphism discriminants remained "out of reach" of the algorithmic $\chi_{18}$ computations underlying Serre's analytic strategy. What remains unresolved is the constructive and methodological completion of this program: a general theory of glueing principally polarized abelian varieties along kernels divisible by the characteristic, and explicit equations for the extremal genus-3 curves whose existence is still established only abstractly, in the generality of Lauter's Theorem 3.

Problem Statement

The starting point is the list of four open issues left by Lauter's 2002 study of extremal genus-3 curves over finite fields. Two of the four are now closed by external evidence (the $p=5$ Diophantine exclusion, closed by the Luca-Tengely-Togbé theorem combined with the reduction $4\cdot 5^e=(2x+1)^2+11$; and the borderline classification of the category equivalence over non-maximal orders, closed by the Jordan-Keeton-Poonen-Rains-Shepherd-Barron-Tate theorem), and the emblematic instance of the explicit-construction thread ($q=243$) is closed by Mestre's effective construction. The surviving open core consists of the following two problems, each open in its own right and independent of the other (neither subsumes the other):

1. Wild-characteristic glueing theory. Develop a general, systematic treatment of glueing principally polarized abelian varieties over finite fields along isogenies whose kernel order is divisible by the characteristic $p$ - so that the kernels are non-étale finite group schemes of mixed étale-local type, as in the $3$-torsion kernels $\mathbb{Z}/3\mathbb{Z}\times\mu_3$ (constant étale part times cube roots of unity) that arise at $q=243$ - replacing the current case-by-case analyses. Such a theory would supply a criterion, expressed in module-theoretic and group-scheme terms, for when a glueing of prescribed polarized factors carries an indecomposable principal polarization in the wild case, with the general criterion required for all prescribed factor glueings (the pair $E_m\times E_m$ and $E_{m-3}$, with product polarizations scaled as required, being one illustrating instance), recovering the known étale (kernel order prime to $p$) theory of Serre and Lauter as the special case.

2. Explicit realizations of abstractly-proven extremal curves. Produce explicit equations (plane quartic or hyperelliptic models, or effective algorithms guaranteed to produce them) for genus-3 curves over $\mathbb{F}_q$ attaining the bounds guaranteed only abstractly by existence arguments - in the generality of Lauter's Theorem 3 (for every $q$, a genus-3 curve with $|N(C)-(q+1)|\ge 3m-3$), beyond the special families already handled: characteristic 2, the $S_3$-family in characteristics 3 and 7, and complex-multiplication discriminants of class number one. This includes extending the algorithmic tools that decide which principally polarized abelian threefolds in a given isogeny class are Jacobians - Serre's obstruction (the non-existence of a suitable indecomposable Hermitian module) on the Hermitian-module side, and the $\chi_{18}$ square criterion certifying non-hyperelliptic Jacobians in characteristic $\ne 2$ on the analytic side - to large endomorphism discriminants.

The two parts are related but independently open. In exceptional fields such as $q=243$, where naive search fails and the glueing kernels $\mathbb{Z}/3\mathbb{Z}\times\mu_3$ have order divisible by the characteristic, explicit constructions would benefit from a wild glueing theory. The coupling is not forced in general, however: Lauter's abstract-existence proof for $q=x^2+j$, $j\in\{1,2\}$ (Theorem 5) glues along étale kernels whose order is prime to the characteristic, and her general-$q$ existence proof (Theorem 3) applies the Serre functor $S$ to indecomposable Hermitian modules rather than glueing, so explicit realizations in the generality of Theorem 3 need not run through a wild glueing theory. Following the separate remarks in the source paper, each part is posed here as an open problem in its own right.

References

1. Kristin Lauter (with an appendix in French by Jean-Pierre Serre), "The maximum or minimum number of rational points on genus three curves over finite fields", Compositio Mathematica 134 (2002), no. 1, 87-111, doi:10.1023/A:1020246226326, https://doi.org/10.1023/A:1020246226326
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