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The maximum number of points on a curve of genus eight over the field of four elements. (English) Zbl 1464.11061

Let \(q\) be a power of a prime number, \(g\) be a non-negative integer, and denote by \(N_{q}(g)\) the maximum number of rational points on a curve of genus \(g\) over the finite field \(\mathbb{F}_{q}\). Studying \(N_{q}(g)\) from both its assimptotic behavior and computation of its actual value is a noteworthy, and often challenging, problem.
For \(q=4\) and \(g=8\), the Oesterlé bound shows that \[ N_{4}(8)\leqslant 24, \] while H. Niederreiter and C. Xing proves in [Acta Arith. 79, No. 1, 59–76 (1997; Zbl 0891.11057)] that there exists a curve of genus \(8\) over the finite field \(\mathbb{F}_4\) with \(21\) rational points.
In this paper, the author improves the previous upper and lower bounds by showing that \[ 22\leqslant N_{4}(8)\leqslant 23, \] where to prove the lower bound, a genus-\(8\) curve over \(\mathbb{F}_{2}\) with \(22\) points over \(\mathbb{F}_{4}\) is constructed as a degree-\(3\) Kummer extension of a genus-\(2\) curve.

MSC:

11G20 Curves over finite and local fields
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry

Citations:

Zbl 0891.11057

Software:

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References:

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