Zeta function of the projective curve \(aY^{2l}= bX^{2l}+ cZ^{2l}\) over a class of finite fileds, for odd primes. (English) Zbl 1066.11027
In a previous article [J. Number Theory 77, No. 2, 288–313 (1999; Zbl 0955.11017)], the author and S. A. Katre found the number of \(\mathbb F_{q^n}\)-rational points on the nonsingular projective curves \(aY^l=bX^l+cZ^l\) and \(aY^{2l}=bX^{2l}+cZ^{2l}\) \((a,b,c\neq 0)\), defined over \(\mathbb F_q\), with \(q=p^\alpha\equiv 1\pmod e\), for \(e=l\) or \(e=2l\), and for odd primes \(l\) and primes \(p\) such that the order of \(l\) modulo \(p\) is even. They also explicitly computed the zeta function of the curve \(aY^l=bX^l+cZ^l\) over \(\mathbb F_q\) in this situation.
Here, the author explicitly computes the zeta function of the curve \(aY^{2l}=bX^{2l}+cZ^{2l}\) over \(\mathbb F_q\) in this situation. There are seven distinct cases depending on the indices of \(b/c\) and \(a/c\) in \(\mathbb F_q^*\). The only such curves that are maximal or minimal curves are the Fermat curves, obtained when \(a=b=c=1\).
Here, the author explicitly computes the zeta function of the curve \(aY^{2l}=bX^{2l}+cZ^{2l}\) over \(\mathbb F_q\) in this situation. There are seven distinct cases depending on the indices of \(b/c\) and \(a/c\) in \(\mathbb F_q^*\). The only such curves that are maximal or minimal curves are the Fermat curves, obtained when \(a=b=c=1\).
Reviewer: Robert F. Lax (Baton Rouge)
MSC:
| 11G20 | Curves over finite and local fields |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14G15 | Finite ground fields in algebraic geometry |
| 14H25 | Arithmetic ground fields for curves |
Citations:
Zbl 0955.11017References:
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