Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves (Q2833089)

Let \(q\) be a power of an odd prime, \(\mathbb F_q\) the field of \(q\) elements and let \(\phi\) denote the quadratic character on \(\mathbb F_q\). The author considers the sums NEWLINE\[NEWLINE\begin{aligned} \psi_{(m,n)}(a,b) &=\sum_{x\in\mathbb F_q} \phi(x^m+a)\phi(x^n+b)\\ \phi_{(m,n)}(a,b) &=\sum_{x\in\mathbb F_q} \phi(x)\phi(x^m+a)\phi(x^n+b), \end{aligned} NEWLINE\]NEWLINE where \(a,b\in\mathbb F_q^*\). When \(q\equiv 1\pmod{2m}\), formulas are given for \(\psi_{(m,m)}\) and \(\phi_{(m,m)}\) in terms of Gaussian hypergeometric series of the shape NEWLINE\[NEWLINE {}_2F_1\begin{pmatrix} A_0 & A_1\\ &B_1\end{pmatrix}\bigg| x =\frac{q}{q-1}\sum_{\chi} \binom{A_0\chi}{\chi}\binom{A_1\chi}{B_1\chi}\chi(x), NEWLINE\]NEWLINE where the sum is over all characters of \(\mathbb F_q\) and NEWLINE\[NEWLINE \binom{A}{B}=\frac{B(-1)}{q}\sum_{x\in\mathbb F_q} A(x)\bar B(1-x). NEWLINE\]NEWLINE There are also formulas for \(\psi_{(2,2)}\) when \(q\equiv 3\pmod4\) and for \(\psi_{(2,4)}\).NEWLINENEWLINEFor the hyperelliptic curves NEWLINE\[NEWLINE C_m: y^2=(x^m+a)(x^m+b)\qquad C_m':y^2=x(x^m+a)(x^m+b) NEWLINE\]NEWLINE the number of \(\mathbb F_q\)-rational points is easily seen to be NEWLINE\[NEWLINE |C_m(\mathbb F_q)|=2+q+\psi_{(m,m)}(a,b)\qquad |C^{\prime}_m(\mathbb F_q)|=1+q+\phi_{(m,m)}(a,b). NEWLINE\]NEWLINE Hence there are expressions for the number of \(\mathbb F_q\)-rational points of \(C_m\) and \(C_m'\) in terms of Gaussian hypergeometric series.