DUAL HOPF ORDERS IN GROUP RINGS OF ELEMENTARY ABELIAN p-GROUPS

Let R be the valuation ring ofK, a finite extension of Qp containing a primitive pth root of unity, and let G be an elementary abelian p-group of order p, with dual group Ĝ. We construct a new family of triangular Hopf orders over R in KG, a proper subfamily whose duals are also triangular, and a proper subfamily of that family whose construction extends the truncated exponential construction of [GC98]. In contrast to the Hopf orders of [GC98],the construction yields examples where none of the rank p subquotients are Larson orders. For G a finite abelian group of order p, the classification of Hopf orders over R in KG is known only for n = 1 [TO70] and 2 [Gr92], [By93], [Un94]. For n > 2 only a few families of Hopf orders were known until recently: [Ra74] for G elementary abelian of order p, [La76] for arbitrary G (”Larson orders”), [Un96] for G cyclic of order p, [CS98] and [GC98] for G elementary abelian of order p. For G of order p, Larson orders are described completely by n valuation parameters (that determine the discriminant of the Hopf order); the examples of [Gr92] and [CS98], [GC98] suggest that a general Hopf order of rank p should involve, in addition, n(n−1)/2 unit parameters, which can conveniently be laid out as entries of a lower triangular matrix. There are two ways that these unit parameters can arise. In the formal group construction of [CS98], the matrix of unit parameters is used to construct an isogeny of polynomial formal groups whose kernel is represented by a Hopf order. The matrix entries then show up as coefficients of group elements in the algebra generators of the Hopf order. In the constructions of [Un96] and [CS98], the unit parameters appear directly in the algebra generators, generalizing Greither’s construction in [Gr92]. This approach was codified in [UC05] in the definition of triangular Hopf orders. This is the fourth in a recent series of papers that construct new families of Hopf orders with the desired number of parameters. [CU03] used formal groups to construct families of Hopf orders in KG when G is cyclic of order p; [UC05] obtained several families of triangular Hopf