Counting rational points of quartic diagonal hypersurfaces over finite fields

1.   Introduction and main results

Let $\mathbb{F}_q$ be the finite field of order $q$ where $q = p^{k}$, $k$ is a positive integer and $p$ is an odd prime. Let $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. For $f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, \cdots, x_n]$, we use $N_q(f; n)$ to denote the number of $\mathbb{F}_q$-rational points of the affine hypersurface $f(x_1, \cdots, x_n) = 0$, namely,

To evaluate the values of $N_q(f; n)$ is a fundamental problem in algebra, number theory and arithmetic geometry. Generally speaking, it is difficult to give explicit formulae for $N_q(f; n)$. An explicit formula for $N_q(f; n)$ is known when $\deg(f)\leq2$ (see ^[13]). Finding the explicit formula for $N_q(f; n)$ under certain conditions has attracted many researchers for many years (see, for instance, ^{[2,3,4,5,6,7,8,9,10,11,14,15,16,17,18,19,20,21]}).

A special diagonal hypersurface over $\mathbb{F}_{q}$ is given by an equation of the type

with $e$ being a positive integer, coefficients $a_1, a_2, \cdots, a_n\in\mathbb{F}_{q}^*$ and $c\in\mathbb{F}_{q}$. It is clear that

For $e = 2$, there is an explicit formula for $N(a_1x_1^2+\cdots+a_nx_n^2 = c)$ in ^[13]. When $q = p^{2t}$ with $p^r\equiv -1({{\rm{mod}}} \ e)$ for a divisor $r$ of $t$ and $e\mid (q-1)$, Wolfmann ^[17] gave an explicit formula of the number of rational points of the hypersurface

over $\mathbb{F}_{q}$ in 1992.

For the special case where $a_1 = a_2 = \cdots = a_n = 1$, we denote by

In 1977, Chowla et al. ^[5] got the generating function $\sum\limits_{n = 1}^{\infty}M^{(3)}_n(0)x^n$ over $\mathbb{F}_{p}$ with $p\equiv1({\rm{mod}} \ 3)$. In 1979, Myerson ^[14] extended the result in ^[5] to the field $\mathbb{F}_{q}$, and also showed that the generating function $\sum\limits_{n = 1}^{\infty}M^{(4)}_n(0)x^n$ over $\mathbb{F}_{q}$ with $p\equiv1({\rm{mod}} \ 4)$ is a rational function in $x$.

In 2020, Zhao et al. ^[19,20] investigated the number of rational points of the hypersurfaces

over $\mathbb{F}_{q}$, with $c\in\mathbb{F}_{q}^*$. For any $c\in\mathbb{F}_q$, in 2022, by using the cyclotomic theory and exponential sums, Zhao et al. ^[21] showed that the generating function $\sum\limits_{n = 1}^{\infty}M^{(4)}_n(c)x^n$ is a rational function in $x$.

In this paper, we consider the problem of finding the explicit formula for the number of rational points of the diagonal quartic hypersurface

over $\mathbb{F}_q$, where $q = p^k$, $a_1, a_2, \cdots, a_n\in\mathbb{F}_q^*$ and $c\in\mathbb{F}_q$.

If $p\equiv 3({\rm{mod}}\ 4)$ and $k$ is an odd integer, then $\gcd(4, q-1) = 2$. It follows that (see ^[12])

Throughout this paper, we let $\eta$ be the quadratic multiplicative character of $\mathbb{F}_q$. Then from Theorems 6.26 and 6.27 in ^[13], the following result is deduced.

Theorem 1.1. Let $q = p^k$ with $p\equiv 3({\rm{mod}}\ 4)$ and $k$ be an odd integer. Let $\psi(c) = -1$ for $c\in\mathbb{F}_q^*$ and $\psi(0) = q-1$. Then the number of rational points of the hypersurface

over $\mathbb{F}_q$ is

if $n$ is even, and is

if $n$ is odd.

If $p\equiv3({\rm{mod}} \ 4)$ and $k$ is an even integer, the following result can be derived from [17, Theorem 1].

Theorem 1.2. Let $p\equiv3({\rm{mod}} \ 4)$ be a prime, $k$ an even integer, $q = p^k$, $s = \frac{q-1}{4}$, $n\ge 2$ and $c\in\mathbb{F}_q$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$. Denote by $N$ the number of rational points of the hypersurface

over $\mathbb{F}_q$, then

if $c = 0$, and

if $c\neq0$, where $v(j)$ is the number of $i$, $1\leq i\leq n$, such that

$\theta(c)$ is the number of $i$, $1\leq i\leq n$, such that $a_i^s = (-c)^s$ and $\tau(j)$ is the number of $i$, $1\leq i\leq n$, such that $a_i^s = (\alpha^j)^s$.

However, the explicit formula for $N(a_1x_1^4+a_2x_2^4+\cdots+a_nx_n^4 = c)$ is still unknown when $p\equiv1({\rm{mod}} \ 4)$. In this paper, we solve this problem by using the Jacobi sums and an analog of the Hasse-Davenport theorem. We give an explicit formula for the number of rational points of diagonal quartic hypersurface

and

over $\mathbb{F}_q$, with $a_1, a_2, b_1, b_2, b_3\in\mathbb{F}_q^*$, $c\in\mathbb{F}_q$ and the characteristic $p\equiv 1({\rm{mod}}\ 4)$. The case with arbitrary $n\ge 4$ variables can be deduced from the reduction formula for Jacobi sums, but we omit the tedious details here.

For a generator $\alpha$ of $\mathbb{F}^*_q$, we define the index of $\beta\in\mathbb{F}^*_q$ with respect to $\alpha$, denoted by $\text{ind}_\alpha\beta$, to be the unique integer $r\in [1, q-1]$ such that $\beta = \alpha^r$ (see, for instance, ^[13]). For any nonzero integer $n$ and prime number $p$, we define $\nu_p(n)$ as the greatest integer $t$ such that $p^t$ divides $n$. Then $\nu_p(n)$ is a nonnegative integer, and $\nu_p(n)\geq$ 1 if and only if $p$ divides $n$.

To give the main results, we need two concepts as follows.

Definition 1.1. Let $\lambda$ be a multiplicative character of $\mathbb{F}_q$. Associated to $\lambda$, we define the function $S_\lambda$ over $\mathbb{F}_q^*$ as follows

Clearly, if $\lambda$ is the multiplicative character of order $4$ of $\mathbb{F}_q$ with $\lambda(\alpha) = i = \sqrt{-1}$, then we have

Definition 1.2. Let $r$, $s$ and $k$ be positive integers, and we define

and

Moreover, let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\beta\in \mathbb{F}_q^*$. Then associated to $r$, $s$ and $k$, we define

Now we can state the main results of this paper as follows.

Theorem 1.3. Let $k$ be a positive integer and $q = p^k$ with $p = 4t+1$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$, $\eta$ be the quadratic multiplicative character of $\mathbb{F}_q$ and $\lambda$ be a multiplicative character of order $4$ of $\mathbb{F}_q$ such that $\lambda(\alpha) = i$. Let $N_1$ denote the number of rational points of the hypersurface over $\mathbb{F}_q$ defined by (1.1). Then

if $c = 0$, and

if $c\neq0$, with $\alpha_1 = c^2a_1a_2$, $\alpha_2 = ca_1^2a_2$, $\alpha_3 = ca_1a_2^2$ and the integers $u$ and $v$ being defined uniquely by

where $\left(\frac{2}{p}\right)$ is the Legendre symbol.

Theorem 1.4. Let $k$ be a positive integer and $q = p^k$ with $p = 4t+1$. Let $\alpha$ be a primitive element of $\mathbb{F}^*_q$, $\eta$ be the quadratic multiplicative character of $\mathbb{F}_q$ and $\lambda$ be a multiplicative character of order $4$ of $\mathbb{F}_q$ such that $\lambda(\alpha) = i$. Let $N_2$ denote the number of rational points of the hypersurface over $\mathbb{F}_q$ defined by (1.2). Then

if $c = 0$, and

if $c\neq0$, with

and the integers $u$ and $v$ being defined as in Theorem 1.3.

This paper is organized as follows. In Section 2, we recall some useful known lemmas that will be needed later. Subsequently, in Section 3, we prove Theorems 1.3 and 1.4. Finally, in Section 4, we supply two examples to illustrate the validity of our results.

2.   Preliminary lemmas

In this section, we present some useful lemmas that are needed in the proof of Theorems 1.3 and 1.4.

Let $q = p^k$, where $k$ is a positive integer and $p$ is a prime. For any element $\beta\in E = \mathbb{F}_{q}$ and $F = \mathbb{F}_p$, the norm of $\beta$ relative to $\mathbb{F}_p$ is defined by (see, for example, ^[1,13])

For the simplicity, we write $\mathbb{N}(\beta)$ for $\mathbb{N}_{E/F} (\beta)$. It is clear that if $\alpha$ is a primitive element of $\mathbb{F}_{q}$, then $\mathbb{N}(\alpha)$ is a primitive element of $\mathbb{F}_{p}$. Let $\chi$ be a multiplicative character of $\mathbb{F}_q$. For any $\alpha\in \mathbb{F}_q$, if $\chi(\alpha) = 1$, then we call the character $\chi$ is trivial. Let $\chi$ be a multiplicative character of $\mathbb{F}_{p}$. Then $\chi$ can be lifted to a multiplicative character $\lambda$ of $\mathbb{F}_{q}$ by setting $\lambda(\beta) = \chi(\mathbb{N}(\beta))$. Any characters of $\mathbb{F}_p$ can be lifted to characters of $\mathbb{F}_q$, but not all the characters of $\mathbb{F}_q$ can be obtained by lifting a character of $\mathbb{F}_p$. The following lemma characterizes all the characters of $\mathbb{F}_q$ that can be obtained by lifting a character of $\mathbb{F}_p$.

Lemma 2.1. ^[1] Let $\mathbb{F}_p$ be a finite field and $\mathbb{F}_q$ be an extension of $\mathbb{F}_p$. A multiplicative character $\lambda$ of $\mathbb{F}_q$ can be lifted by a multiplicative character $\chi$ of $\mathbb{F}_p$ if and only if $\lambda^{p-1}$ is trivial.

Let $\lambda_1$, $\cdots$, $\lambda_n$ be $n$ multiplicative characters of $\mathbb{F}_q$. The Jacobi sum $J(\lambda_1, \cdots, \lambda_n)$ is defined by

where the summation is taken over all $n$-tuples $(\gamma_1, \cdots, \gamma_n)$ of elements of $\mathbb{F}_q$ with $\gamma_1+\cdots+\gamma_n = 1$. It is clear that if $\sigma$ is a permutation of $\{1, \cdots, n\}$, then

The readers are referred to ^[1,13] for the basic facts on the Jacobi sum.

The following lemma is an analog of the Hasse-Davenport theorem which establishes an important relationship between the Jacobi sums in $\mathbb{F}_q$ and the Jacobi sums in $\mathbb{F}_p$.

Lemma 2.2. ^[13] Let $\chi_1, \cdots, \chi_n$ be multiplicative characters of $\mathbb{F}_p$, not all of which are trivial. Suppose $\chi_1, \cdots, \chi_n$ are lifted to characters $\lambda_1, \cdots, \lambda_n$, respectively, of the extension field $\mathbb{F}_q$ of $\mathbb{F}_p$ with degree $[\mathbb{F}_q:\mathbb{F}_p] = k$. Then

Lemma 2.3. (Reduction formula for the Jacobi sums) ^[1] Let $\lambda_1, \cdots, \lambda_{s-1}, \lambda_s$ be $s$ nontrivial multiplicative characters of $\mathbb{F}_q$. If $s\geq2$, then

Lemma 2.4. ^[1] Let $p\equiv 1({\rm{mod}} \ 4)$ be a prime, $q$ a power of $p$, $\alpha$ be a generator of $\mathbb{F}_q^*$, and let $\chi$ be a multiplicative character of order 4 of $\mathbb{F}_p$ with $\chi(\mathbb{N}(\alpha)) = i$. Then

where the integers $u$ and $v$ are uniquely determined by

Lemma 2.5. ^[1] Let $p = 4t+1$ be a prime number. Let $g$ be a primitive element of $\mathbb{F}_p$ and $\chi$ be a multiplicative character of order $4$ over $\mathbb{F}_p$ such that $\chi(g) = i$. Let the integers $u$ and $v$ be defined as in Lemma 2.4. Then the values of the $16$ Jacobi sums $J(\chi^m, \chi^n)$ $(m, n = 0, 1, 2, 3)$ of order $4$ are given in Table 1.

The following lemma gives a formula for the number of rational points of a diagonal hypersurface in terms of the Jacobi sums.

Lemma 2.6. ^[1] Let $k_1, \cdots, k_n$ be positive integers. Let $a_1, \cdots, a_n\in\mathbb{F}_{q}^*$ and $c\in\mathbb{F}_{q}$. Set

and let $\lambda_i$ be a multiplicative character of $\mathbb{F}_{q}$ of order $d_i$, for $i = 1, \cdots, n$. Then the number $N$ of rational points of the equation

over $\mathbb{F}_{q}$ is given by

if $c = 0$, and by

if $c\neq 0$.

3.   Proof of Theorems 1.3 and 1.4

In this section, we give the proofs of Theorems 1.3 and 1.4. First, we begin with a lemma.

Lemma 3.1. Let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be a multiplicative character of order $4$ of $\mathbb{F}_q = \mathbb{F}_{p^k}$ such that $\lambda(\alpha) = i$. Then for any positive integers $r$, $s$ and $\beta\in\mathbb{F}_q^*$, we have

where the function $W_{(r, s, k)}(\beta)$ is defined as in Definition 1.2.

Proof. If ${\rm ind}_{\alpha}\beta\equiv 0 ({\rm{mod}}\ 4)$, then

Thus, one has

If ${\rm ind}_{\alpha}\beta\equiv 1 ({\rm{mod}}\ 4)$, then

Thus, one has

If ${\rm ind}_{\alpha}\beta\equiv 2 ({\rm{mod}}\ 4)$, then

and if ${\rm ind}_{\alpha}\beta\equiv 3 ({\rm{mod}}\ 4)$, then

The results in these two cases can be proved similarly. □

We can now give the proof of Theorem 1.3.

Proof of Theorem 1.3. Let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be a multiplicative character of $\mathbb{F}_q$ of order 4 with $\lambda(\alpha) = i$. Since $q\equiv 1({\rm{mod}} \ 4)$, then

Using Lemma 2.6, by setting $\lambda_1 = \lambda_2 = \lambda$, one can deduce that the number $N_1$ of rational points

in $\mathbb{F}_q^2$ is given by

if $c = 0$, and by

if $c\neq 0$.

Since $p\equiv 1({\rm{mod}} \ 4)$, it follows that $\lambda^{p-1}$ is trivial. Thus, from Lemma 2.1, we know that the quartic multiplicative character $\lambda$ can be lifted by a quartic multiplicative character $\chi$ of $\mathbb{F}_p$.

Using Lemmas 2.2, 2.5 and 3.1 and Definition 1.1, we have the following two cases, depending on $c = 0$ or $c\ne 0$.

If $c = 0$, we derive that

Thus, from (3.1) and (3.3), the first part of Theorem 1.3 follows immediately.

If $c\neq0$, we obtain

Thus, from (3.2) and (3.4), the desired result follows immediately. □

Now, we can turn our attention to prove Theorem 1.4.

Proof of Theorem 1.4. By the same argument as in the proof of Theorem 1.3, let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be the multiplicative character of $\mathbb{F}_q$ of order 4 with $\lambda(\alpha) = i$. One has

if $c = 0$, and

if $c\neq 0$.

Clearly, the quartic multiplicative character $\lambda$ can be lifted by a quartic multiplicative character $\chi$ of $\mathbb{F}_p$. Thus, from Lemmas 2.2, 2.3, 2.5 and 3.1 and Definition 1.2, one gets that

Then, from (3.5) and (3.7), the first part of Theorem 1.4 follows immediately.

We can now turn our attention to prove the second part of Theorem 1.4. Clearly,

By Lemmas 2.5 and 3.1 and Definition 1.2, we derive that

Using Lemma 2.5, one has

Thus, from (3.6), (3.8)–(3.10) and Definition 1.1, the desired result of the second part of Theorem 1.4 follows immediately. □

4.   Examples

In this section, we present two examples to demonstrate the validity of Theorems 1.3 and 1.4. We have validated these two examples by using the Magma which is a powerful algebraic computation program package.

Example 4.1. Let $p = 5$ and $k = 5$. It can be checked easily that $2$ is a primitive element of $\mathbb{F}_5$. Choose a primitive element $\omega$ of $\mathbb{F}_{5^5}$ with $\mathbb{N}(\omega) = 2$. We consider the numbers of rational points $(x_1, x_2)\in\mathbb{F}^2_{5^5}$ of the quartic hypersurfaces

over $\mathbb{F}_{5^5}$.

Now, the Legendre symbol

Thus, the integers $u$ and $v$ are determined by

Then, one has $u = 1, \ v = 2.$ Thus, by Theorem 1.3, we obtain

Example 4.2. Let $p = 13$ and $k = 2$. We know that $2$ is a primitive element of $\mathbb{F}_{13}$. Choose a primitive element $\omega$ of $\mathbb{F}_{13^2}$ with $\mathbb{N}(\omega) = 2$. We consider the numbers of rational points $(x_1, x_2, x_3)\in\mathbb{F}^3_{13^2}$ of the quartic hypersurfaces

over $\mathbb{F}_{13^2}$.

Now, the Legendre symbol

Thus, the integers $u$ and $v$ are determined by

Therefore,

By Theorem 1.4, we have

5.   Conlusions

Studying the number of rational points of the polynomial equation

over $\mathbb{F}_{q}$ is a fundamental problem in algebra, number theory and arithmetic geometry. Generally speaking, it is difficult to give an explicit formula for the number of solutions of the equation

There are many researchers who concentrated on finding the formula for the number of solutions of

under certain conditions. Exponential sums are important tools for solving problems involving the number of solutions of the equation

over $\mathbb{F}_{q}$. In this paper, by using the Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for

and

with

and $c\in \mathbb{F}_q$. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface

of $n\geq 4$ variables with

and $p\equiv1({\rm{mod}} \ 4)$, can also be deduced.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the National Science Foundation of China (Nos. 12026223, 12026224 and 12231015), the National Key Research and Development Program of China (No. 2018YFA0704703), the Natural Science Foundation of Beijing (No. M23017), the Natural Science Foundation of Henan Province (No. 232300420123) and the Research Center of Mathematics and Applied Mathematics, Nanyang Institute of Technology.

Conflict of interest

We declare that we have no conflicts of interest.