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05182615 Lachaud, Gilles The Klein quartic as a cyclic group generator. Mosc. Math. J. 5, No. 4, 857-868 (2005). 2005 Independent University of Moscow, Moscow; American Mathematical Society, Providence, RI EN varieties over finite fields arithmetic varieties curves over arithmetic ground fields Jacobians rational points zeta functions cryptography Zbl 0126.07001 Zbl 0369.14011 Zbl 0991.11032 Let $k$ be an arbitrary field, and let $a$, $b$, $c$ be three elements in $k^X$. Then the equation $ax^3y+ by^3z+ cz^3x= 0$ defines a nonsingular projective plane curve $X(a,b,c)$ of degree 4 and genus 3 which specializes to the famous classical Klein quartic when $a= b= c= 1$, and the Jacobian $J_{X(a,b,c)}$ of $X(a,b,c)$ appears as a three-dimensional Abelian variety defined over $k$. In the paper under review, the author studies the curves $X(a,b,c)$ and their Jacobians in the arithmetic case where $k=\bbfF_q$ is a finite field of characteristic $p$ such that $q\equiv 1\pmod n$ for a given prime number $n$. Extending some earlier work by {\it D. K. Faddeev} [Tr. Mat. Inst. Steklova 64, 284--293 (1961; Zbl 0126.07001)], {\it H. Gross} and {\it D. E. Rohrlich} [Invent. Math. 44, 201--224 (1978; Zbl 0369.14011)], and {\it N. D. Elkies} [Math. Sci. Res. Inst. Publ. 35, 51--101 (1999; Zbl 0991.11032)], he first derives isomorphic models for the curves $X(a,b,c)$ and shows how those a related, via nonramified Galois coverings, to the Fermat curve defined by the equation $ax^7+ by^7+ cz^7= 0$. Applying this to the special case where $k= \bbfF_q$ with $q\equiv 1\pmod 7$, the author determines then the numerator of the zeta function of the quartic $X(a,b,c)$ explicitly, which finally is used to conclude some explicit formulas for the number of points of the group $J_{X(a,b,c)}(k)$ of rational points in the Jacobian of the quartic $X(a,b,c)$. In particular, if $k=\bbfF_p$ is a prime field, then these numerical results imply that $J_{X(a,b,c)}(\bbfF_p)$ is a cyclic group of prime order for a significant number of choices of the parameters. As the author points out, these cyclic groups of type $J_{X(a,b,c)}(\bbfF_p)$ might be very useful for cryptographic applications. Werner Kleinert (Berlin)