<p>We introduce the notion of a relative of the Hermitian curve of degree <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sqrt{q}+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msqrt> <mi>q</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>, which is a plane curve defined by <Equation ID="Equ32"> <EquationSource Format="TEX">$\begin{aligned} (x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A \, ^t \!(x,y,z) =0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <msqrt> <mi>q</mi> </msqrt> </msup> <mo>,</mo> <msup> <mi>y</mi> <msqrt> <mi>q</mi> </msqrt> </msup> <mo>,</mo> <msup> <mi>z</mi> <msqrt> <mi>q</mi> </msqrt> </msup> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> <mmultiscripts> <mspace width="0.166667em" /> <mrow /> <mi>t</mi> </mmultiscripts> <mspace width="-0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A \in GL(3, \mathbb {F}_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and we study their basic properties. One of the basic properties is that the number of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-points of any relative of the Hermitian curve of degree <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sqrt{q}+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msqrt> <mi>q</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is congruent to 1 modulo <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sqrt{q}$</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>q</mi> </msqrt> </math></EquationSource> </InlineEquation>. In the latter part of this paper, we classify those curves having two or more rational inflexions.</p>

Relatives of the Hermitian curve

期刊论文
发表日期：

2026年1月28日

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Masaaki Homma,
Seon Jeong Kim

摘要

We introduce the notion of a relative of the Hermitian curve of degree $\sqrt{q}+1$ $\sqrt{q} + 1$ over $\mathbb {F}_q$ $F_{q}$ , which is a plane curve defined by $\begin{aligned} (x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A \, ^t \!(x,y,z) =0 \end{aligned}$ $\begin{matrix} (x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}}) A^{t} (x, y, z) = 0 \end{matrix}$ with $A \in GL(3, \mathbb {F}_q)$ $A \in G L (3, F_{q})$ , and we study their basic properties. One of the basic properties is that the number of $\mathbb {F}_q$ $F_{q}$ -points of any relative of the Hermitian curve of degree $\sqrt{q}+1$ $\sqrt{q} + 1$ is congruent to 1 modulo $\sqrt{q}$ $\sqrt{q}$ . In the latter part of this paper, we classify those curves having two or more rational inflexions.