The a-numbers of Fermat and Hurwitz curves

Abstract

For an algebraic curve X defined over an algebraically closed field of characteristic p > 0, the a-number a(X ) is the dimension of the space of exact holomorphic differentials on X . We compute the a-number for an infinite families of Fermat and Hurwitz curves. Our results apply to Hermitian curves giving a new proof for a previous result of Gross [9].

Automorphisms of generalized Fermat curves

“…In the particular case that X is a plane curve, a formula for C was given in [24]. This formula has been used in [18] to compute a X for the case of classical Fermat curves (and also some Hurwitz curves), in [4] for certain quotients of Ree curves, in [11] for the Hermitian curve and in [7] for the case of Suzuki curves. As, for generalized Fermat curves (which are not longer planar models) we have obtained an explicit basis, we hope they can be used to describe their exact holomorphic forms.…”

On $BT_1$ group schemes and Fermat Jacobians

“…1.3]. An analogue of parts (1), (3), and (4) below for F d was proven by Montanucci and Speziali [MS18] by a direct calculation of the Cartier operator. 8.4.…”

“…In Section 8, we detour to record some combinatorial preliminaries, and we recover several known results about Fermat curves, cf. [Yui80], [KW88], [Gon97], and [MS18]. We hope that seeing these results in a unified framework will be useful to the reader.…”

The $a$-number of Certain Hyperelliptic Curves

“…The a-number of Hermitian curves computed by Gross in [10], and for Fermat and Hurwitz curves computed by Maria [14]. A few results on the rank of the Carteir operator (especially a-number) of curves introduced by Kodama and Washio [11], Gonzlez [8], Pries and Weir [15] and Yui [22].…”