Isomorphism class

Abstract: In this paper we study the moduli space A0 Fp of polarized abelian varieties of dimensiongin positive characteristic. We construct a stratification of this space. The strata are indexed by isomorphism classes of group schemes killed byp;a polarized abelian variety (X, A) has its moduli point in a certain stratum if X [p] belongs to the isomorphism class given by a certain discrete invariant. We define these invariants by a numerical property of a filtration ofN= X [p].Passing from one stratum to a stratum in its boundary feels like “degenerating the p-structure”. The fact that these strata are all quasi-affine allows us to keep going in this process until we arrive at the unique zero-dimensional stratum, the superspecial locus. One can formulate this idea by saying that the ordinary locus has several “boundaries”, one where the abelian variety degenerates, one where the p-structure “becomes more special” (and an analogous idea for all non-zero-dimensional strata). This phenomenon, non-present in this form in characteristic zero, but available and powerful in positive characteristic, is ex-pected to have many applications.

Abstract: We explore the topological full group [[G]] of an essentially principal etale groupoid G on a Cantor set. When G is minimal, we show that [[G]] (and its certain normal subgroup) is a complete invariant for the isomorphism class of the etale groupoid G. Furthermore, when G is either almost finite or purely infinite, the commutator subgroup D([[G]]) is shown to be simple. The etale groupoid G arising from a onesided irreducible shift of finite type is a typical example of a purely infinite minimal groupoid. For such G, [[G]] is thought of as a generalization of the Higman-Thompson group. We prove that [[G]] is of type F∞, and so in particular it is finitely presented. This gives us a new infinite family of finitely presented infinite simple groups. Also, the abelianization of [[G]] is calculated and described in terms of the homology groups of G.

Abstract: In this paper we study abelian varieties and p-divisible groups in characteristic P Even though a non-trivial deformation of an abelian variety can produce a non trivial Galois-representation, say on the Tate-?-group of the generic fiber (in any characteristic), the geometric generic fiber has a "constant" Tate-^-group. We en counter the same phenomenon for the p-structure in positive characteristic: any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p-divisible groups. However, for non-ordinary abelian varieties this seems to break down. The fas cinating structure which comes out of this is that the maximal locus where a given geometric isomorphism class of a p-divisible group is realized (e.g., in a family of abelian varieties) is a locally closed set. The locus defined by the geometric iso morphism type of a p-divisible group will be called a "central leaf"; see 2.1 and 3.4. This gives rise to a "foliation" of the open stratum attached to a Newton polygon ?; the dimension of any "leaf" in the same Newton polygon stratum solely depends on ?. In extreme cases either the leaf is the whole stratum, as in the ordinary case, or in the "almost ordinary case" (the p-rank equals g ? 1), or a leaf is zero-dimensional as in the supersingular case; in intermediate cases a leaf can be a proper subset and still be positive dimensional: we have worked out the example of g = 4 in 8.1.

Abstract: In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the \(\ell \)-isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring’s correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras.