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Plane curves with many points over finite fields. (English) Zbl 1066.14025

Let \(k=\mathbb F_p\) be the finite field of prime order \(p>2\) and \(f(x,y)\) a polynomial in \(k[x,y]\) of degree \(d<p\). Suppose that \(f(x,y)\) does not have a linear component defined over \(k\). In [Proc. Lond. Math. Soc., III Ser. 52, 1–19 (1986; Zbl 0593.14020)] K.-O. Stöhr and J. F. Voloch noticed that the number of solutions of the curve \(f(x,y)=0\) in \(k^2\) is upper bounded by \(\Delta:=d(p+d-1)/2\), provided that \(f(x,y)\) is absolutely irreducible and \(f\) does not divide \(f_{xx}(f_y)^2-2f_{xy}f_xf_y+f_{yy}(f_x)\). In the paper under review, the authors show that the same bound is true for any \(f\) satisfiyng the linearity condition. Moreover, the curve is nonsingular if equality holds.
For \(p\equiv 1\pmod{1}\), examples of curves attaining \(\Delta\) can be obtained by slicing the surface \(w^{(p-1)/2}+y^{(p-1)/2}-z^{(p-1)/2}-x^{(p-1)/2}=0\) by the plane \(w=y+cz\) with \(c\) a nonsquare in \(\mathbb F_p\). Such a surface has many points for its degree; see [J. F. Voloch, Contemp. Math. 324, 219–226 (2003; Zbl 1040.11046)]. For \(p\equiv 3\mod{4}\) the authors also point out explicit examples of curves attaining \(\Delta\). In particular, the case \(p=11\) and \(d=4\) give an example of an optimal curve of genus 3 over \(\mathbb F_{11}\) which previously was noticed by Serre via a not explicit construction. By general methods one can improve \(\Delta\) for \(d<p/15\) or \(d\geq p\). The authors ask whether or not the bound \(\Delta\) is attained for \(p/15\leq d <p\).

MSC:

14G15 Finite ground fields in algebraic geometry
11G20 Curves over finite and local fields
11G05 Elliptic curves over global fields
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[1] K. Lauter (with an appendix by J-P. Serre), The maximum or minimum number of rational points on curves of genus three over finite fields , 2001. · Zbl 0982.14015
[2] F. Rodríguez Villegas and J.F. Voloch, On certain plane curves with many integral points , Experimental Math. 8 (1999), 57-62. · Zbl 1029.11025 · doi:10.1080/10586458.1999.10504388
[3] F. Rodríguez Villegas, J.F. Voloch and D. Zagier, Constructions of plane curves with many points , Acta Arith. 99 (2001), 85-96. · Zbl 1042.11039 · doi:10.4064/aa99-1-8
[4] K.-O. Stöhr and J.F. Voloch, Weierstrass points and curves over finite fields , Proc. London Math. Soc. 52 (1986), 1-19. · Zbl 0593.14020 · doi:10.1112/plms/s3-52.1.1
[5] J.F. Voloch, Surfaces in \(\mathbf P^3\) over finite fields , in Topics in Algebraic and Noncommutative Geometry: Proc. in Memory of Ruth Michler (C. Melles et al. eds.), Contemp. Math., vol. 324, Amer. Math. Soc., Providence, 2003, pp. 219-226. · Zbl 1040.11046
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