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Optimal curves over finite fields with discriminant \(-19\). (English) Zbl 1219.11094

Summary: We study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant \(-19\). We prove that any such curve can be given by an explicit equation of certain form (see Theorem 5.1). Using these equations we obtain a table of maximal and minimal curves over prime finite fields with discriminant \(-19\) of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.

MSC:

11G20 Curves over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25 Arithmetic ground fields for curves
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[1] Deligne, Pierre, Variétés abéliennes ordinaires sur un corps fini, Invent. Math., 8, 238-243 (1969) · Zbl 0179.26201
[2] Kani, E.; Rosen, M., Idempotent relations and factors of Jacobians, Math. Ann., 284, 2, 307-327 (1989) · Zbl 0652.14011
[3] Lauter, Kristin, The maximum or minimum number of rational points on genus three curves over finite fields, Compos. Math., 134, 1, 87-111 (2002), with an appendix by Jean-Pierre Serre · Zbl 1031.11038
[4] Oort, Frans; Ueno, Kenji, Principally polarized abelian varieties of dimension two or three are Jacobian varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 20, 377-381 (1973) · Zbl 0272.14008
[5] van der Geer, Gerard; van der Vlugt, Marcel, Supersingular curves of genus 2 over finite fields of characteristic 2, Math. Nachr., 159, 73-81 (1992) · Zbl 0774.14045
[6] Schiemann, Alexander, Classification of Hermitian forms with the neighbour method, J. Symbolic Comput., 26, 4, 487-508 (1998) · Zbl 0936.68129
[7] Waterhouse, William C., Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4), 2, 521-560 (1969) · Zbl 0188.53001
[8] Zaytsev, A., Optimal curves of low genus over finite fields (2007)
[9] Ritzenthaler, Christophe, Explicit computations of Serreʼs obstruction for genus 3 curves and application to optimal curves, LMS J. Comput. Math., 13, 192-207 (2010) · Zbl 1278.11068
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