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Pointless curves of genus three and four

Everett W. Howe·Kristin E. Lauter·Jaap Top
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Abstract

A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field F_q if and only if q < 26, that there exist pointless smooth plane quartics over F_q if and only if either q < 24 or q = 29 or q = 32, and that there exist pointless genus-4 curves over F_q if and only if q < 50.

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Everett W. Howe, Kristin E. Lauter, Jaap Top. 2004-03-10. Pointless curves of genus three and four. https://arxiv.org/abs/math/0403178

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