Further results on rational points of the curve $\displaystyle y^{q^n}-y=\gamma x^{q^h+1} - \alpha$ over ${\mathbb {F}}_{q^m}$

Abstract

Let q be a positive power of a prime number. For arbitrary positive integers h, n, m with n dividing m and arbitrary $\gamma ,\alpha \in {\mathbb {F}}_{q^m}$ with $\gamma \ne 0$ the number of ${\mathbb {F}}_{q^m}$-rational points of the curve $y^{q^n}-y=\gamma x^{q^h+1} - \alpha$ is determined in many cases (Özbudak and Saygı, in: Larcher et al. (eds.) Applied algebra and number theory, 2014) with odd q. In this paper we complete some of the remaining cases for odd q and we also present analogous results for even q.

Appendix

Here we recall some of the results obtained in [5] for completeness. Let p be odd and q, m, h, n, $\alpha$, $\gamma$, A, N be defined as above in Sect. 1.

Let N(m, n) denote the cardinality

$$\begin{aligned} N(m,n) = \left| \left\{ x \in {\mathbb {F}}_{q^m} \mid \mathrm{Tr}_{{\mathbb {F}}_{q^{m}}/{\mathbb {F}}_{q^n}}\left( \gamma x^{q^h+1} -\alpha \right) = 0 \right\} \right| . \end{aligned}$$

Therefore, we have $N=q^n N(m,n)$. The number N(m, n) is computed in [5] instead of N.

Theorem 5

Assume that $s\le t$. Let $\eta$ and $\eta '$ denote the quadratic characters of ${\mathbb {F}}_q$ and ${\mathbb {F}}_{q^m}$, respectively.

• If m / n is even and $A=0$, then

  $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} - \left( q^n-1\right) q^{m/2-n} &{} \quad \text{ if }\, \eta \left( (-1)^{m/2}\right) \eta '(\gamma )=1, \\ q^{m-n} + \left( q^n-1\right) q^{m/2-n} &{} \quad \text{ if }\, \eta \left( (-1)^{m/2}\right) \eta '(\gamma )=-1. \end{array} \right. \end{aligned}$$

• If m / n is even and $A\ne 0$, then

  $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} + q^{m/2-n} &{}\quad \text{ if } \eta \left( (-1)^{m/2}\right) \eta '(\gamma )=1, \\ q^{m-n} - q^{m/2-n} &{} \quad \text{ if } \eta \left( (-1)^{m/2}\right) \eta '(\gamma )=-1. \end{array} \right. \end{aligned}$$

• If m / n is odd and $A=0$, then

  $$\begin{aligned} N(m,n)=q^{m-n}. \end{aligned}$$

• If m / n is odd, $A\ne 0$ and n is even, then

  $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} + q^{(m-n)/2} &{} \quad \text{ if } (u_1,u_2) \in \left\{ (1,1), (-1,-1)\right\} , \\ q^{m-n} - q^{(m-n)/2} &{}\quad \text{ if } (u_1,u_2) \in \left\{ (1,-1), (-1,1)\right\} , \end{array} \right. \end{aligned}$$

  where $u_1$ and $u_2$ are the integers in the set $\{-1,1\}$ given by

  $$\begin{aligned} u_1=\eta \left( (-1)^{m/2} \right) \eta '(\gamma ) \text{ and } u_2=\eta \left( (-1)^{n/2}\right) \eta '(A). \end{aligned}$$

• If m / n is odd, $A\ne 0$ and n is odd, then

  $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} + q^{(m-n)/2} &{} \quad \text{ if }\, (u_1,u_2) \in \left\{ (1,1), (-1,-1)\right\} , \\ q^{m-n} - q^{(m-n)/2} &{}\quad \text{ if }\, (u_1,u_2) \in \left\{ (1,-1), (-1,1)\right\} , \end{array} \right. \end{aligned}$$

  where $u_1$ and $u_2$ are the integers in the set $\{-1,1\}$ given by

  $$\begin{aligned} u_1=\eta \left( (-1)^{(m-1)/2} \right) \eta '(\gamma ) \;\text{ and }\;u_2=\eta \left( (-1)^{(n-1)/2}\right) \eta '(A). \end{aligned}$$

Theorem 6

Assume that $s \ge t+1$ and $u\le t$. Let $\omega$ be a generator of the multiplicative group ${\mathbb {F}}_{q^m} \setminus \{0\}$ and let a be the integer with $0 \le a < q^m-1$ such that $\gamma =\omega ^a$.

• Case $s=t+1$ Put $q_1=q^{2^tr}$.

  • If $a \not \equiv m_1 \frac{q_1+1}{2} \mod (q_1+1)$, then

    $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} + q^{m/2-n} &{}\quad \text{ if }\, A \ne 0, \\ q^{m-n} -(q^n-1)q^{m/2-n} &{} \quad \text{ if }\, A =0. \end{array} \right. \end{aligned}$$

  • If $a \equiv m_1 \frac{q_1+1}{2} \mod (q_1+1)$, then for $k=2^{t+1}r$ we have that

    $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} - q^{(m+k)/2-n} &{} \quad \text{ if }\,A \ne 0, \\ q^{m-n} + (q^n-1)q^{(m+k)/2-n} &{} \quad \text{ if }\,A =0. \end{array} \right. \end{aligned}$$

• Case $s \ge t+2$ Put $q_1=q^{2^tr}$.

  • If $a \not \equiv 0 \mod (q_1+1)$, then

    $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} - q^{m/2-n} &{}\quad \text{ if }\,A \ne 0, \\ q^{m-n} + (q^n-1)q^{m/2-n} &{}\quad \text{ if }\,A =0. \end{array} \right. \end{aligned}$$

  • If $a \equiv 0 \mod (q_1+1)$, then for $k=2^{t+1}r$ we have that

    $$\begin{aligned} N(m,n)=\left\{ \begin{array}{ll} q^{m-n} + q^{(m+k)/2-n} &{} \quad \text{ if }\,A \ne 0, \\ q^{m-n} - (q^n-1)q^{(m+k)/2-n} &{} \quad \text{ if }\,A =0. \end{array} \right. \end{aligned}$$

Theorem 7

Assume that $t+1\le u \le s$ and $A=0$. Let $\omega$ be a generator of the multiplicative group ${\mathbb {F}}_{q^m} \setminus \{0\}$ and let a be the integer with $0 \le a < q^m-1$ such that $\gamma =\omega ^a$.

• Case $s=t+1$ Put $B_1=\gcd \left( m_2, q^{2^t \rho }+1\right)$.

  • If $a \equiv n_1 m_2 \frac{q^{2^t r}+1}{2} \mod \left( \frac{q^{2^t r}+1}{q^{2^t \rho }+1}B_1\right)$, then

    $$\begin{aligned} N(m,n)=q^{m-n} - (q^{n}-1)q^{m/2-n} + B_1\frac{q^n-1}{q^{2^t \rho }+1}\left( q^{m/2 + 2^t r -n} + q^{m/2-n}\right) . \end{aligned}$$

  • If $a \not \equiv n_1 m_2 \frac{q^{2^t r}+1}{2} \mod \left( \frac{q^{2^t r}+1}{q^{2^t \rho }+1}B_1\right)$, then

    $$\begin{aligned} N(m,n)=q^{m-n} - (q^{n}-1)q^{m/2-n}. \end{aligned}$$

• Case $s\ge t+2$ Put $B_1=\gcd \left( 2^{s-u} m_2, q^{2^t \rho }+1\right)$.

  • If $a \not \equiv n_1 m_2 \frac{q^{2^t r}+1}{2} \mod \left( \frac{q^{2^t r}+1}{q^{2^t \rho }+1}B_1\right)$, then

    $$\begin{aligned} N(m,n)=q^{m-n} - (q^{n}-1)q^{m/2-n}. \end{aligned}$$

  • If $a \equiv 0 \mod \left( \frac{q^{2^t r}+1}{q^{2^t \rho }+1}B_1\right)$, then

    $$\begin{aligned} N(m,n)=q^{m-n} + (q^{n}-1)q^{m/2-n} - B_1\frac{q^n-1}{q^{2^t \rho }+1}\left( q^{m/2 + 2^t r -n} + q^{m/2-n}\right) . \end{aligned}$$

  • If $a \not \equiv 0 \mod \left( \frac{q^{2^t r}+1}{q^{2^t \rho }+1}B_1\right)$, then

    $$\begin{aligned} N(m,n)=q^{m-n} + (q^{n}-1)q^{m/2-n}. \end{aligned}$$