The 𝑎-numbers of Jacobians of Suzuki curves

Abstract

For m ∈ N, let Sm be the Suzuki curve defined over F 2 2m+1 . It is well-known that Sm is supersingular, but the p-torsion group scheme of its Jacobian is not known. The a-number is an invariant of the isomorphism class of the p-torsion group scheme. In this paper, we compute a closed formula for the a-number of Sm using the action of the Cartier operator on H 0 (Sm, Ω 1 ).Let m ∈ N, q = 2 2m+1 , and q 0 = 2 m . The Suzuki curve S m ⊂ P 2 is defined over F q by the homogeneous equation:This curve is smooth and irreducible with genus g = q 0 (q − 1) and it has exactly one point at infinity [8, Proposition 1.1]. The number of points on the Suzuki curve over F q is #S m (F q ) = q 2 + 1; this number is optimal in that it reaches Serre's improvement to the Hasse-Weil bound [8, Proposition 2.1].In fact, S m is the unique F q -optimal curve of genus g [2]. This shows that S m is the Deligne-Lusztig variety of dimension 1 associated with the group Sz(q) = 2 B 2 (q) [7, Proposition 4.3]. The curve S m has the Suzuki group Sz(q) as its automorphism group; the order of Sz(q) is q 2 (q − 1)(q 2 + 1) which is very large compared with g. Because of the large number of rational points relative to their genus, the Suzuki curves provide good examples of Goppa codes [4, Section 4.3], [5], [8].The L-polynomial of S m is (1 + √ 2qt + qt 2 ) g [7, Proposition 4.3]. It follows that S m is supersingular for each m ∈ N. This fact implies that the Jacobian Jac(S m ) is isogenous to a product of supersingular elliptic curves and that Jac(S m ) has no 2-torsion points over F 2 . However, there are still open questions about Jac(S m ). In this paper, we address one of these by computing a closed formula for the a-number of Jac(S m ).The a-number is an invariant of the 2-torsion group scheme Jac(S m )[2]. Specifically, if α 2 denotes the kernel of Frobenius on the additive group G a , then the a-number of S m is a(m) = dim F2 Hom(α 2 , Jac(S m )[2]). It equals the dimension of the intersection of Ker(F ) and Ker(V ) on the Dieudonné module of Jac(S m )[2]. Having a supersingular Newton polygon places constraints upon the a-number but does not determine it. The a-number also gives partial information about the decomposition of Jac(S m ) into indecomposable principally polarized abelian varieties, Lemma 4.3, and about the Ekedahl-Oort type of Jac(S m )[2], see Section 4.2.In Section 4, we prove that the a-number of S m is a(m) = q 0 (q 0 + 1)(2q 0 + 1)/6, see Theorem 4.1. The proof uses the action of the Cartier operator on H 0 (S m , Ω 1 ) as computed in Section 3.

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.…”

The de Rham cohomology of the Suzuki curves

“…The 2-torsion group scheme Jac(S m ) [2] is a BT 1 -group scheme of rank 2 2gm . In [7], the authors show that the a-number of Jac(S m ) [2] is a m = q 0 (q 0 + 1)(2q 0 + 1)/6; in particular, lim m→∞ a m /g m = 1/6. However, the Ekedahl-Oort type of Jac(S m ) [2] is not known.…”

On the Zeta function and the automorphism group of the generalized Suzuki curve

“…Furthermore, in [4] and [22], the investigation of genera and plane models for quotients of the Suzuki curve is addressed. Past studies also examined embeddings in P N [2], [11], class field theory [28], the invariant a-number [15], and Weierstrass semigroups and coding theory [3], [8], [12], [14], [20], [27], [31]. More recently, the construction of a Galois cover of the Suzuki curve in [37] raised a number of other issues to be investigated [19], [33].…”