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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.
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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