We conjecture a deformation of the Weyl character formula for type G 2 in the spirit of Tokuyama's formula for type A. Using our conjecture, we prove a combinatorial version of the Gindikin-Karpelevič formula for G 2 , in the spirit of Bump-Nakasuji's formula for type A.
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.
Abstract. We study the structure of p-parts of Weyl group multiple Dirichlet series. In particular, we extend results of Chinta, Friedberg, and Gunnells [6] and show, in the stable case, that the p-parts of Chinta and Gunnells [7] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [2,3]. In this vein, we give an explicit recurrence relation on the coefficients of the p-parts, which allows us to describe the support of the p-parts and address the extent to which they are uniquely determined.
Let N be a connected and simply connected nilpotent Lie group, and let K be a subgroup of the automorphism group of N . We say that the pair (K, N ) is a nilpotent Gelfand pair if L 1 K (N ) is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs (K, N ) where the K-orbits in the center of N have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set ∆(K, N ) of bounded K-spherical functions on N and the set A(K, N ) of K-orbits in the dual n * of the Lie algebra for N established in [BR08] is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for N a free group and N a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs in [BR08].
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