Small complete arcs and caps in Galois spaces over finite fields Fq\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_q$$\end{document} with characteristic greater than three are constructed from singular cubic curves. For m\documentclass[12pt]{minimal}
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\begin{document}$$m$$\end{document} a divisor of q+1\documentclass[12pt]{minimal}
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\begin{document}$$q+1$$\end{document} or q-1\documentclass[12pt]{minimal}
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\begin{document}$$q-1$$\end{document}, complete plane arcs of size approximately q/m\documentclass[12pt]{minimal}
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\begin{document}$$q/m$$\end{document} are obtained, provided that (m,6)=1\documentclass[12pt]{minimal}
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\begin{document}$$(m,6)=1$$\end{document} and m<14q1/4\documentclass[12pt]{minimal}
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\begin{document}$$m<\frac{1}{4}q^{1/4}$$\end{document}. If in addition m=m1m2\documentclass[12pt]{minimal}
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\begin{document}$$m=m_1m_2$$\end{document} with (m1,m2)=1\documentclass[12pt]{minimal}
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\begin{document}$$(m_1,m_2)=1$$\end{document}, then complete caps in affine spaces of dimension N≡0(mod4)\documentclass[12pt]{minimal}
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\begin{document}$$N\equiv 0 \pmod 4$$\end{document} with roughly m1+m2mqN/2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{m_1+m_2}{m}q^{N/2}$$\end{document} points are described. These results substantially widen the spectrum of q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}s for which complete arcs in AG(2,q)\documentclass[12pt]{minimal}
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\begin{document}$$AG(2,q)$$\end{document} of size approximately q3/4\documentclass[12pt]{minimal}
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\begin{document}$$q^{3/4}$$\end{document} can be constructed. Complete caps in AG(N,q)\documentclass[12pt]{minimal}
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\begin{document}$$AG(N,q)$$\end{document} with roughly q(4N-1)/8\documentclass[12pt]{minimal}
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\begin{document}$$q^{(4N-1)/8}$$\end{document} points are also provided. For infinitely many q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}s, these caps are the smallest known complete caps in AG(N,q)\documentclass[12pt]{minimal}
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\begin{document}$$AG(N,q)$$\end{document}, N≡0(mod4)\documentclass[12pt]{minimal}
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\begin{document}$$N \equiv 0 \pmod 4$$\end{document}.
