Ihara-maximal curves: uniqueness of the genus-7 curve over $\mathbb{F}_7$, and existence of Ihara-maximal curves of genus 19 over $\mathbb{F}_{19}$ and genus 24 over $\mathbb{F}_{16}$

For a curve $X$ of genus $g$ over the finite field $\mathbb{F}_q$, Ihara's bound refines the Hasse-Weil bound: $\#X(\mathbb{F}_q)\le q+1+(\sqrt{(8q+1)g^2+(4q^2-4q)g}-g)/2$. Aubry, Herbaut and Monaldi proved that a curve attaining this bound (an Ihara-maximal curve) has all its Frobenius eigenvalues given by a single conjugate pair, forcing its zeta function to $(1+aT+qT^2)^g/((1-T)(1-qT))$ and its Jacobian to be isogenous to a power of a single simple abelian variety; they classified all Ihara-maximal curves of genus $g\le 18$ (up to $\mathbb{F}_q$-isomorphism) and all those over fields with $q\le 13$, except for the uniqueness of the genus-7 curve over $\mathbb{F}_7$; beyond these ranges, the first open cases are genus 19 over $\mathbb{F}_{19}$ and genus 24 over $\mathbb{F}_{16}$. This problem asks to settle those three cases: the uniqueness up to $\mathbb{F}_7$-isomorphism of the known Ihara-maximal curve of genus 7 over $\mathbb{F}_7$ with 36 rational points; the existence of a curve of genus 19 over $\mathbb{F}_{19}$ with $\#X(\mathbb{F}_{19})=\#X(\mathbb{F}_{19^2})=153$; and the existence of a curve of genus 24 over $\mathbb{F}_{16}$ with $\#X(\mathbb{F}_{16})=\#X(\mathbb{F}_{16^2})=161$. Resolving them completes the classification for all genera $g\le 19$ and all prime powers $q\le 16$. In each case the Jacobian is forced to be isogenous to a power of one simple abelian variety - the $g$-th power of an ordinary elliptic curve in the first two cases, and necessarily a higher-dimensional variety in the third, since no elliptic curve over $\mathbb{F}_{16}$ has Frobenius trace $-6$ - and no known bound or elimination technique decides the case. All three cases are open: the gain of the 2025 Serre-type refinement of the Ihara bound vanishes at exactly these parameter points.

Background

Let $\mathbb{F}_q$ be the finite field with $q$ elements ($q$ a prime power) and let $X$ be a smooth, projective, absolutely irreducible curve of genus $g$ over $\mathbb{F}_q$; write $N_n=\#X(\mathbb{F}_{q^n})$ for the number of points rational over the degree-$n$ extension. The Hasse-Weil bound $N_1\le q+1+2g\sqrt{q}$ was improved by Ihara for large genus to

$N_1-(q+1)\le\frac{\sqrt{(8q+1)g^2+(4q^2-4q)g}-g}{2},$

which is sharper than the Weil bound once $g>\sqrt{q}(\sqrt{q}-1)/2$. A curve attaining equality in Ihara's inequality is called Ihara-maximal; the genus of such a curve is at most $\sqrt{q}(q-1)/\sqrt{2}$ (a bound of Hallouin and Perret quoted by the primary source), which makes the classification problem finite for each $q$. Curves attaining the Weil bound itself (possible only when $q$ is a square) are called Weil-maximal; a classical theorem of Rück and Stichtenoth identifies the Weil-maximal curve of maximal genus $\sqrt{q}(\sqrt{q}-1)/2$ as the Hermitian curve $x^{\sqrt{q}+1}+y^{\sqrt{q}+1}+z^{\sqrt{q}+1}=0$. The counting function $N_q(g)=\max \#X(\mathbb{F}_q)$, the maximum over all curves of genus $g$ over $\mathbb{F}_q$, is tabulated with its best known upper and lower bounds in the community database manYPoints.

The zeta function of $X$ is $Z_X(T)=L_X(T)/((1-T)(1-qT))$, where the $L$-polynomial $L_X(T)=\prod_{i=1}^{2g}(1-\omega_i T)$ encodes the Frobenius eigenvalues $\omega_i$, algebraic numbers of absolute value $\sqrt{q}$ determined by the action of the $q$-power Frobenius map on the cohomology of $X$; point counts are recovered as $N_n=q^n+1-\sum_i \omega_i^n$. The Jacobian $J_X$ of $X$ is the $g$-dimensional abelian variety parametrizing degree-zero divisor classes on $X$; its characteristic polynomial is $\prod_i (T-\omega_i)$. Two abelian varieties are isogenous when they are linked by a surjective homomorphism with finite kernel, and isogenous abelian varieties have the same characteristic polynomial. Hallouin and Perret proved the bound $N_2-(q^2+1)\le 2gq-\frac{1}{g}(N_1-(q+1))^2$ from non-negativity of a Gram determinant built from the Frobenius correspondence; curves attaining equality in it are called Hallouin-Perret-maximal (HP-maximal).

Aubry, Herbaut and Monaldi gave the structure theory behind these bounds: the gap in the Hallouin-Perret inequality equals $4g$ times the variance of the real parts of the Frobenius eigenvalue pairs, and a curve is Ihara-maximal if and only if it is both HP-maximal and Diophantine-stable, meaning $\#X(\mathbb{F}_{q^2})=\#X(\mathbb{F}_q)$. Consequently an Ihara-maximal curve has all Frobenius eigenvalues equal to one conjugate pair, so its zeta function is forced to the form

$Z_X(T)=\frac{(1-2\alpha T+qT^2)^g}{(1-T)(1-qT)}$

with $2\alpha=(q+1-N_1)/g$ an integer, and its Jacobian is isogenous to a power of a single $\mathbb{F}_q$-simple abelian variety $A$ whose characteristic polynomial is a power of $T^2-2\alpha T+q$ (Proposition 2.5 of the primary source). By the Deuring-Waterhouse classification of elliptic-curve isogeny classes (as recorded in Remark 2.8 of the primary source), the simple factor $A$ is an elliptic curve $E/\mathbb{F}_q$ with Frobenius trace $2\alpha$ - so that the Jacobian is isogenous to the power $E^g$ - exactly when the trace $2\alpha$ satisfies one of the Deuring-Waterhouse conditions; in particular this holds, with $E$ ordinary (that is non-supersingular), whenever $\gcd(2\alpha,p)=1$, where $p$ is the characteristic of $\mathbb{F}_q$. This covers the cases $(g,q)=(7,7)$ and $(19,19)$ below (traces $-4$ and $-7$); at $(g,q)=(24,16)$ the trace is $-6$, which satisfies none of the conditions over $\mathbb{F}_{16}=\mathbb{F}_{2^4}$ (the even traces occurring there are only $0$ and $\pm4,\pm8$), so no elliptic curve over $\mathbb{F}_{16}$ has trace $-6$ and $A$ has dimension $d>1$ with $d\mid 24$ - in fact $d>2$, since the case analysis behind Theorem 5.1 of the primary source shows that $(T^2+6T+16)^2$ is not the characteristic polynomial of any abelian surface over $\mathbb{F}_{16}$. Equivalently, the real Weil polynomial - the degree-$g$ polynomial whose roots are the conjugate-pair sums $\omega_i+q/\omega_i$, here all equal to $2\alpha$ - is the perfect power $(x-2\alpha)^g$.

On this basis Aubry, Herbaut and Monaldi enumerated the finitely many pairs $(g,q)$ for which the Ihara bound is an integer (the quantity $(8q+1)g^2+4qg(q-1)$ must be a perfect square) and is not excluded by any known upper bound on $N_q(g)$, and constructed or identified a curve for each surviving pair. Their conclusion, recorded in their Table 1: the complete list of Ihara-maximal curves of genus $g\le 18$ up to $\mathbb{F}_q$-isomorphism consists of ten curves at nine couples $(g,q)$ - three elliptic curves ($y^2+y=x^3+x$ over $\mathbb{F}_2$ with 5 points, $y^2=x^3+2x+1$ over $\mathbb{F}_3$ with 7 points, and the Hermitian elliptic curve $x^3+y^3+z^3=0$ over $\mathbb{F}_4$ with 9 points); two genus-3 curves over $\mathbb{F}_4$ with 14 points each, and the Fermat curve $x^4+y^4+z^4=0$ of genus 3 over $\mathbb{F}_9$ with 28 points; the Fermat (Hermitian) curve $x^5+y^5+z^5=0$ of genus 6 over $\mathbb{F}_{16}$ with 65 points; a genus-7 curve over $\mathbb{F}_7$ with 36 points; the Fermat curve $x^6+y^6+z^6=0$ of genus 10 over $\mathbb{F}_{25}$ with 126 points; and, in genus 14 over $\mathbb{F}_8$, the Suzuki curve $y^8-y=x^2(x^8-x)$ with 65 points, identified via a maximality theorem of Fuhrmann and Torres that they quote - the genus-7 entry being the only one whose completeness (uniqueness up to $\mathbb{F}_q$-isomorphism) is undetermined. Over the fields with $q\le 13$ the same classification holds with the $q=16$ and $q=25$ entries removed: eight curves at the seven couples $(1,2)$, $(1,3)$, $(1,4)$, $(3,4)$ (two $\mathbb{F}_4$-isomorphism classes), $(3,9)$, $(7,7)$ and $(14,8)$. Three frontier cases remain. First, at $(g,q)=(7,7)$ an Ihara-maximal curve exists - $N_1=N_2=36$, $L_X(T)=(1+4T+7T^2)^7$, realized by the fibre product of two cyclic triple covers of the projective line found by Özbudak, Gülmez Temür and Yayla - but uniqueness up to $\mathbb{F}_7$-isomorphism is undetermined. (A Kummer cover is a cyclic cover obtained by extracting an $m$-th root of a rational function; a fibre product of two such covers over the same base line is a curve mapping to both.) Second, at $(g,q)=(19,19)$, the first genus beyond their method, the existence of a curve with $N_1=N_2=153$ and zeta function $(1+7T+19T^2)^{19}/((1-T)(1-19T))$ is open. Third, at $(g,q)=(24,16)$, the first field beyond $q\le 13$ (since 14 and 15 are not prime powers), the existence of a curve with $N_1=N_2=161$ and zeta function $(1+6T+16T^2)^{24}/((1-T)(1-16T))$ is open. According to the manYPoints table, $N_7(7)=36$ exactly (lower bound equal to upper bound), $N_{19}(19)$ has no submitted lower bound at all and upper bound 153 (the Ihara bound), and $N_{16}(24)$ lies in the interval $[129,161]$. The uneliminated real Weil polynomials $(x+4)^7$, $(x+7)^{19}$ and $(x+6)^{24}$ are integral and satisfy all standard arithmetic constraints, and the toolbox for deciding whether an isogeny class contains a Jacobian - the standard route to eliminating such polynomials - is developed mainly for abelian surfaces (dimension 2), far from dimensions 7, 19 and 24.

Problem Statement

Settle the three named frontier cases that remain in the classification of Ihara-maximal curves over finite fields:

(i) Uniqueness at $(g,q)=(7,7)$. Is there, up to $\mathbb{F}_7$-isomorphism, exactly one Ihara-maximal curve of genus 7 over $\mathbb{F}_7$? Equivalently: is every smooth projective absolutely irreducible curve of genus 7 over $\mathbb{F}_7$ with 36 rational points $\mathbb{F}_7$-isomorphic to the fibre product of the cyclic triple covers $y_1^3=5(x+2)(x+5)/x$ and $y_2^3=3x^2(x+5)/(x+3)$ of the projective line (the known example, with zeta function $(1+4T+7T^2)^7/((1-T)(1-7T))$)?

(ii) Existence at $(g,q)=(19,19)$. Does there exist a smooth projective absolutely irreducible curve of genus 19 over $\mathbb{F}_{19}$ with $N_1=N_2=153$? Equivalently: does there exist a curve of genus 19 over $\mathbb{F}_{19}$ whose zeta function is $(1+7T+19T^2)^{19}/((1-T)(1-19T))$, i.e. whose Jacobian is isogenous to the 19-th power of the elliptic curve over $\mathbb{F}_{19}$ of Frobenius trace $-7$?

(iii) Existence at $(g,q)=(24,16)$. Does there exist a smooth projective absolutely irreducible curve of genus 24 over $\mathbb{F}_{16}$ with $N_1=N_2=161$? Equivalently: does there exist a curve of genus 24 over $\mathbb{F}_{16}$ whose zeta function is $(1+6T+16T^2)^{24}/((1-T)(1-16T))$? (Since no elliptic curve over $\mathbb{F}_{16}$ has Frobenius trace $-6$, such a Jacobian would be isogenous to a power of a higher-dimensional simple abelian variety with characteristic polynomial a power of $T^2+6T+16$.)

These three determinations are the named remaining gaps of one classification program and share one completion boundary: settling (i) completes the 'complete list up to isomorphism' of Ihara-maximal curves of genus $g\le 18$; settling (ii) extends the classification to all genera $g\le 19$, since $(19,19)$ is the only pair $(19,q)$ for which the Ihara bound is an integer within the admissible genus range (so the elimination procedure of the primary source leaves only this couple at genus 19); and settling (iii) extends it to all prime powers $q\le 16$, since the couples at $q=16$ for which the Ihara bound is an integer are $(6,16)$ (realized by the Hermitian curve), $(10,16)$ (eliminated by the known bound $N_{16}(10)\le 86<87$) and $(24,16)$, so $(24,16)$ is the only pair left open at $q=16$. A complete answer resolves all three cases, each either by an explicit construction or by a proof of non-existence (respectively uniqueness), as the mathematical truth demands.

References

1. Yves Aubry, Fabien Herbaut, Julien Monaldi, 'Maximal curves with respect to quadratic extensions over finite fields', International Journal of Number Theory (2026), pp. 1-27. DOI: 10.1142/S1793042126501101; arXiv:2506.08603 (v3, 26 May 2026). URL: https://arxiv.org/abs/2506.08603 . (Primary source.)
2. Yasutaka Ihara, 'Some remarks on the number of rational points of algebraic curves over finite fields', Journal of the Faculty of Science, University of Tokyo, Section IA Mathematics 28(3) (1981), pp. 721-724. No DOI exists for this journal.
3. Emmanuel Hallouin, Marc Perret, 'A unified viewpoint for upper bounds for the number of points of curves over finite fields via euclidean geometry and semi-definite symmetric Toeplitz matrices', Transactions of the American Mathematical Society 372(8) (2019), pp. 5409-5451. DOI: 10.1090/tran/7813. URL: https://doi.org/10.1090/tran/7813
4. Ferruh Özbudak, Burcu Gülmez Temür, Oğuz Yayla, 'An exhaustive computer search for finding new curves with many points among fibre products of two Kummer covers over F_5 and F_7', Turkish Journal of Mathematics 37(6) (2013), pp. 908-913. DOI: 10.3906/mat-1206-26. URL: https://doi.org/10.3906/mat-1206-26
5. Emmanuel Hallouin, Philippe Moustrou, Marc Perret, 'Refinements on higher order Weil-Oesterlé bounds via a Serre type argument', Bulletin of the London Mathematical Society 58(9) (published online 31 August 2026); arXiv:2506.05212 (v1, 5 June 2025). DOI: 10.1112/blms.70483. URL: https://arxiv.org/abs/2506.05212
6. manYPoints - table of curves with many points over finite fields (edited by G. van der Geer, E. W. Howe, K. Lauter, C. Ritzenthaler). URL: https://manypoints.org
7. Hans-Georg Rück, Henning Stichtenoth, 'A characterization of Hermitian function fields over finite fields', Journal für die reine und angewandte Mathematik (Crelle's Journal) 457 (1994), pp. 185-188. DOI: 10.1515/crll.1994.457.185. URL: https://doi.org/10.1515/crll.1994.457.185