On Maximal Curves and Unramified Coverings

We discuss sufficient conditions for a given curve to be covered by a maximal curve with the covering being unramified; it turns out that the given curve itself will be also maximal. We relate our main result to the question of whether or not a maximal curve is covered by the Hermitian curve. We also provide examples illustrating the results. §1. Let X be a projective, geometrically irreducible, non-singular algebraic curve of genus g defined over the finite field Fq2 with q 2 elements, and let J be the Jacobian variety of X . A celebrated theorem of A. Weil gives in particular the following upper bound for the cardinality of the set X (Fq2) of Fq2-rational points on the curve: #X (Fq2) ≤ (q + 1) 2 + q(2g − 2) . The curve X is called Fq2-maximal if it attains the upper bound above; i.e., if one has #X (Fq2) = (q + 1) 2 + q(2g − 2) . Equivalently, the curve X is Fq2-maximal if the Fq2-Frobenius endomorphism on the Jacobian J acts as a multiplication by −q; see [13], [6, Lemma 1.1]. The most well-known example of a Fq2-maximal curve is the so-called Hermitian curve H, which can be given by the plane model X + Y q+1 + Z = 0 . By a result of Serre, see [10, Prop. 6], every curve Fq2-covered by a Fq2-maximal curve is also Fq2-maximal. Thus many examples of Fq2-maximal curves arise by considering quotient curves H/G, where G is a subgroup of the automorphism group of H; see [7], [4], [5]. Serre’s result points out the following. Question. Is any Fq2-maximal curve Fq2-covered by the Hermitian curve H? So far, this question is an open problem but a positive answer is known whenever the genus is large enough: Lemma. ([9, Thm. 3.1], [13], [6, Thm. 3.1], [1]) A Fq2-maximal curve of genus larger than ⌊(q − q + 4)/6⌋ is Fq2-covered by the Hermitian curve. 2000 Mathematics Subject Classification: Primary 11G20; Secondary 14G05, 14G15.