Distributions of traces of Frobenius for smooth plane curves over finite fields. (English) Zbl 1431.14047
Let \(k\) be a finite field of order \(q\). For \(C\) a projective, non-singular,
geometrically irreducible algebraic curve over \(k\) of genus \(g\), a basic problem is to
compute \(N:=\# C(k)\), the number of \(k\)-rational points of \(C\). We have that \(N=q+1-t\),
where \(t\) is the trace of the Frobenius map acting on the first étale cohomology group
\(H^1(J_{\text{ét}},{\mathbb Z}_\ell)\) of the Jacobian \(J\) of \(C\) being \(\ell\) a prime different
of the characteristic of \(k\); moreover, \(|t|\leq g\lfloor 2\sqrt{q}\rfloor\) (the
Hasse-Weil-Serre bound). One is often interesting in computing \(N_q(g)\), the maximal
possible number of \(k\)-rational points on curves of genus \(g\).
J.-P. Serre [Sémin. Théor. Nombres, Univ. Bordeaux I 1982–1983, Exp. No. 22, 8 p. (1983; Zbl 0538.14016)] computed \(N_q(1)\) and \(N_q(2)\), and consider the possibility to obtain a similar result for \(N_q(3)\). He noticed that this task is more involved as here we have the so-called Serre obstruction, namely there are Jacobians of genus 3 curves over \(\bar k\) which are not the Jacobian of curves over \(k\); a reason for this obstruction is the existence of non-hyperelliptic curves of genus bigger than \(2\). Although there have been many computational and conceptual approaches to Serre’s obstruction (see e.g. T. Ibukiyama [Tohoku Math. J. (2) 45, No. 3, 311–329 (1993; Zbl 0819.14007)], G. Lachaud and C. Ritzenthaler [Ser. Number Theory Appl. 5, 88–115 (2008; Zbl 1151.14321)]), there is no so far a closed formula for \(N_q(g)\), \(g\geq 3\).
In the paper under review, among other things, \(N_q(3)\) is investigated via the possible values of the trace of Frobenius: for large negative value of \(t\) it is shown that there are more curves with trace \(t\) than with trace \(-t\). Indeed, let \({\mathcal N}_{q,3}(t)\) be the number of non-hyperelliptic genus \(3\) curves over \(k\) of trace \(t\) up to \(k\)-isomorphism. Then the function \({\mathcal M}_{q,3}(t):={\mathcal N}_{q,3}(t)-{\mathcal N}_{q,3}(-t)\) is considered for \(0\leq t\leq 6\sqrt{q}\). It is shown that \({\mathcal M}_{q,3}(t)\leq 0\) for \(t\) large enough. For instance this is enough to show that there is a genus \(3\)-curve \(C\) over \(k\) such that \(\#C(k)\geq q+1+3\lfloor\sqrt{q}\rfloor-3\) (cf. K. Lauter and J.-P. Serre [Compos. Math. 134, No. 1, 87–111 (2002; Zbl 1031.11038)]). See R. Lercier et al. [LMS J. Comput. Math. 17A, 128–147 (2014; Zbl 1333.14060)] for computations on prime fields of small order.
J.-P. Serre [Sémin. Théor. Nombres, Univ. Bordeaux I 1982–1983, Exp. No. 22, 8 p. (1983; Zbl 0538.14016)] computed \(N_q(1)\) and \(N_q(2)\), and consider the possibility to obtain a similar result for \(N_q(3)\). He noticed that this task is more involved as here we have the so-called Serre obstruction, namely there are Jacobians of genus 3 curves over \(\bar k\) which are not the Jacobian of curves over \(k\); a reason for this obstruction is the existence of non-hyperelliptic curves of genus bigger than \(2\). Although there have been many computational and conceptual approaches to Serre’s obstruction (see e.g. T. Ibukiyama [Tohoku Math. J. (2) 45, No. 3, 311–329 (1993; Zbl 0819.14007)], G. Lachaud and C. Ritzenthaler [Ser. Number Theory Appl. 5, 88–115 (2008; Zbl 1151.14321)]), there is no so far a closed formula for \(N_q(g)\), \(g\geq 3\).
In the paper under review, among other things, \(N_q(3)\) is investigated via the possible values of the trace of Frobenius: for large negative value of \(t\) it is shown that there are more curves with trace \(t\) than with trace \(-t\). Indeed, let \({\mathcal N}_{q,3}(t)\) be the number of non-hyperelliptic genus \(3\) curves over \(k\) of trace \(t\) up to \(k\)-isomorphism. Then the function \({\mathcal M}_{q,3}(t):={\mathcal N}_{q,3}(t)-{\mathcal N}_{q,3}(-t)\) is considered for \(0\leq t\leq 6\sqrt{q}\). It is shown that \({\mathcal M}_{q,3}(t)\leq 0\) for \(t\) large enough. For instance this is enough to show that there is a genus \(3\)-curve \(C\) over \(k\) such that \(\#C(k)\geq q+1+3\lfloor\sqrt{q}\rfloor-3\) (cf. K. Lauter and J.-P. Serre [Compos. Math. 134, No. 1, 87–111 (2002; Zbl 1031.11038)]). See R. Lercier et al. [LMS J. Comput. Math. 17A, 128–147 (2014; Zbl 1333.14060)] for computations on prime fields of small order.
Reviewer: Fernando Torres (Campinas)
MSC:
| 14Q05 | Computational aspects of algebraic curves |
| 14H10 | Families, moduli of curves (algebraic) |
| 14H25 | Arithmetic ground fields for curves |
| 14H37 | Automorphisms of curves |
| 14H45 | Special algebraic curves and curves of low genus |
| 14H50 | Plane and space curves |
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