On holomorphic polydifferentials in positive characteristic

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m‐(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; we give bases for each case when the base field is rational, introduce the Boseck invariants and give an elementary approach to the G module structure of Ω(m) in terms of Boseck invariants. The last computation is achieved without any restriction on the base field in the cyclic case, while in the elementary abelian case it is assumed that the base field is rational. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

Adjoint selmer groups as Iwasawa modules

We fix a primep. In this paper, starting from a given Galois representation ? having values inp-adic points of a classical groupG, we study the adjoint action of ? on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(?) of ?. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(?))_/L for each number fieldL. We scrutinize the behavior of Sel(Ad(?))_{/E ∞} as an Iwasawa module for a fixed ? _p -extensionE _∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ?. WhenG=GL(2), ? is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.     

Automorphism groups of positive entropy on minimal projective varieties

We determine the geometric structure of a minimal projective threefold having two ‘independent and commutative’ automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X,G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fitting the ‘boundary case’.     

Cube structures and intersection bundles

We define the notion of a hypercube structure on a functor between two commutative Picard categories which generalizes the notion of a cube structure on a G_m-torsor over an abelian scheme. We prove that the determinant functor of a relative scheme X/S of relative dimension n is canonically endowed with a (n+2)-cube structure. We use this result to define the intersection bundle I_X/S(L₁,…,L_n+1) of n+1 line bundles on X/S and to construct an additive structure on the functor I_X/S:PIC(X/S)ⁿ⁺¹→PIC(S). Then, we construct the resultant of n+1 sections of n+1 line bundles on X, and the discriminant of a section of a line bundle on X. Finally we study the relationship between the cube structures on the determinant functor and on the discriminant functor, and we use it to prove a polarization formula for the discriminant functor.     

Holomorphic differentials of certain solvable covers of the projective line over a perfect field

We provide a Boseck‐type basis of the space of holomorphic differentials for a large class of solvable covers of the projective line with perfect field of constants of characteristic . Within this class, we also describe the Galois module structure of holomorphic differentials for abelian covers.     

Wildly ramified covers with large genus

We study wildly ramified G-Galois covers branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer σ with p?σ, there exists a G-Galois étale cover of the affine line with conductor σ above the point ∞.     

Involutions of Type G<Subscript>2</Subscript> Over Fields of Characteristic Two

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G₂ over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G₂ over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.     

Group-theoretic constraints on the structure of the class group

Let $\text{K}\text{k}$ be a Galois extension of number fields and G its Galois group. By considering the class group of K as a G module we are able to make assertions about its structure once the class number is known. Applications are made to cyclic cubic fields and the 2-class group of cyclotomic fields.     

The spherical spectrum of a graded ring

We introduce a functor Sph, the spherical spectrum, which assigns to a graded ringG a space Sph(G) of homogeneous orderings ofG. It combines ideas of concrete geometry in theN-sphere defined by positively homogeneous polynomial equations and inequalities with the abstract notion of the real spectrum of a ring to give a counterpart for real semialgebraic geometry of the functor Proj.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pⁿ?Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p³.     

Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties

Let G be a reductive group defined over a p-adic local field L, let P be a parabolic subgroup of G with Levi quotient M, and write G:=G(L), P:=P(L), and M:=M(L). In this paper we construct a functor JP from the category of essentially admissible locally analytic G-representations to the category of essentially admissible locally analytic M-representations, which we call the Jacquet module functor attached to P, and which coincides with the usual Jacquet module functor of [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] on the subcategory of admissible smooth G-representations. We establish several important properties of this functor.     

Mackey and Frobenius Structures on Odd Dimensional Surgery Obstruction Groups

C. T. C. Wall formulated surgery-obstruction groups L _n (Z[G]) in terms of quadratic modules and automorphisms. C. B. Thomas showed that the Wall-group functors L _n (Z[–],w|_–) are modules over the Hermitian-representation-ring functor G₁(Z, –) if the orientation homomorphism w is trivial. A. Bak generalized the notion of quadratic module by introducing quadratic-form parameters, and obtained various K-groups related to quadratic modules and automorphisms. One of the authors established that some Bak groups W _n (Z[G], w) are equivariant-surgery-obstruction groups and showed in the case of even dimension n that the Bak-group functor W _n (Z)[–], _–; w|_–) is a w-Mackey functor as well as a module over the Grothendieck–Witt-ring functor GW₀(Z, –), where w is possibly nontrivial. In this paper, we prove the same facts in the case of odd dimension n.     

Space curves with the expected postulation

The postulation of a space curve is a classifying invariant which computes for any integer n the dimension of the family of surfaces of degree n containing the curve. We prove that for any integers d and g satisfying d−3?g?2d−9, there exists a smooth connected curve of degree d and genus g with the minimal postulation expected by the Riemann-Roch theorem.     

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

Neighborhood of an Embedded <Emphasis Type="Italic">J</Emphasis>-Holomorphic Disc

We study the deformation space of an embedded J-holomorphic disc D in an almost complex surface (X,J). Every such J is shown to be equivalent to a small deformation of a certain model structure J _β along D, where β : D→ℂ is a complex valued function whose modulus |β| is a biholomorphic invariant. Furthermore, we find a nonlinear invertible operator mapping the space of all small J-holomorphic deformations of the given J-holomorphic disc onto the space of small holomorphic deformations of the standard disc in ℂ².     

Flow prolongation of some tangent valued forms

We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frölicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.