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Arithmetic on a family of Picard curves. (English) Zbl 1101.14043
Mullen, Gary L. (ed.) et al., Finite fields with applications to coding theory, cryptography and related areas. Proceedings of the 6th international conference on finite fields and applications, Oaxaca, México, May 21–25, 2001. Berlin: Springer (ISBN 3-540-43961-7/hbk). 187-208 (2002).
The authors consider the family of Picard curves $$C_a: \, Y^3=X^4-aX$$ over a perfect field $$k$$ of characteristic different from 3, where $$a\in k^*$$. When $$k$$ is a finite field, they give expressions for the number of rational points of $$C_a$$ in terms of Jacobi sums associated to certain powers of the ninth power residue character. If $$k$$ is an algebraic number field containing a primitive ninth root of unity $$\zeta_9$$, they show that the $$L$$-function of $$C_a$$ is the inverse of a product of six Hecke $$L$$-functions with Grössencharakter. Over the complex numbers, they review some results of the first author on Picard curves [Geometry and arithmetic. Around Euler partial differential equations. Mathematics and its applications (East European Series), 11 (1986; Zbl 0595.14016); The ball and some Hilbert problems. Lect. Math., ETH Zürich (1995; Zbl 0905.14013)], and show that, up to isomorphism, $$C_1$$ is the only Picard CM-curve with a cyclotomic maximal order, namely $${\mathbb Z}[\zeta_9]$$, as endomorphism ring. The first author, along with J. Estrada-Sarlabous, J.-P. Cherdieu, and E. Reinaldo-Barreiro, has also studied this family of curves [in: VIth Symp. Math., Int. Conf. CIMAF 2001. Proc. Fourth Italian-Latin American Conf. Appl. Ind. Math., Havana, Cuba, March 19-23, 2001, 266–275 (2001; Zbl 1012.14009)].
For the entire collection see [Zbl 0995.00009].

MSC:
 14H25 Arithmetic ground fields for curves 11G20 Curves over finite and local fields 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14H45 Special algebraic curves and curves of low genus 14G35 Modular and Shimura varieties
Keywords:
Picard curve; $$L$$-function; Jacobi sum