Curves of large genus covered by the hermitian curve

Abstract

For the Hermitian curve H dened over the nite eld F q 2 , we give a complete classication of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves F q 2 -covered by H arising from subgroups of the special linear group SL (2; F q ) or from subgroups in the normaliser of the Singer group of the projective unitary group P G U (3; F q 2 ). Since curves F q 2 -covered by H are maximal over F q 2 , our results provide some classication and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2 , both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the \third largest genus problem" including some new results and a detailed account of current w ork is given. MIRAMARE { TRIESTE

On additive polynomials and certain maximal curves

“…As J.P. Serre has shown, a subcover of a maximal curve is maximal (see [10]). So one way to construct explicit maximal curves is to find equations for Galois subcovers of the Hermitian curve (see [3,7]). …”

On the genus of a maximal curve

“…+ y 2 + y = x q+1 . These results together with some evidence coming from [12], [13], [19] make it plausible that only few F q 2 -maximal curves can have genus close to the upper limit q(q − 1)/2. As a matter of fact, in the range…”

“…Remark 4. F q 2 -maximal curves of genus (q 2 − q + 4)/6 do exist as the following examples show, see[19],[13, Thm. 2.1]:…”

The Möbius function of PSU(3, 2^{2^n})

“…The number of elements of order 3 stabilizing a fixed triangleT is 2(q 2 − q + 1), because any element in M 4 (T ) \ C q 2 −q+1 has order 3 (see [4,Section 4]).…”