A note on lattices in semi-stable representations

Let p be a prime, K a finite extension over \mathbb Q_p{{\mathbb Q}_p} and G = Gal([`(K)] /K){G = {\rm Gal}(\overline K /K)} . We extend Kisin’s theory on j{\varphi} -modules of finite E(u)-height to give a new classification of G-stable \mathbb Z1_p{{\mathbb Z}1_p} -lattices in semi-stable representations.     

Jacobians and differents of projective varieties

Let X be a reduced and irreducible projective variety of dimension d. Let π:X→Y be a separable noetherian normalization of X and ? the canonical morphism Ω^d _X/k→Ω^d _L/k. From our main result: $$J_X \varphi (\pi ^* \Omega ^d _{Y/k} ) = \theta _k (X/Y)\varphi (\Omega ^d _{X/k} )$$ we deduce relations among: complementary module C(X/Y), Kähler different θ_k(X/Y), Dedekind different θ_D(X/Y), jacobian ideal J_K and ω-jacobian ideal \(\tilde J_X\) .     

Degeneration of weight spectral sequences

 A notion of morphism of semi-stable type is a higher dimensional analogue of semi-stable degeneration over the unit disc. For a proper surjective morphism of semi-stable type, the author constructed a cohomological mixed Hodge complex which gives a candidate of the limit of Hodge structures. In this article we define finite increasing filtrations on the cohomological mixed Hodge complex above and prove the E ₂ -degeneracy of the spectral sequences obtained from these filtrations. Received: 5 December 2000 / Revised version: 29 January 2002     

Semi-stable models for rigid-analytic spaces

 Let R be a complete discrete valuation ring with field of fractions K and let X _K be a smooth, quasi-compact rigid-analytic space over S_p K. We show that there exists a finite separable field extension K' of K, a rigid-analytic space X' _K' over S_p K' having a strictly semi-stable formal model over the ring of integers of K', and an étale, surjective morphism f : X' _K' →X _K of rigid-analytic spaces over S_p K. This is different from the alteration result of A.J. de Jong [dJ] who does not obtain that f is étale. To achieve this property we have to work locally on X _K , i.e. our f is not proper and hence not an alteration. Received: 26 October 2001 / Revised version: 14 August 2002 Published online: 14 February 2003     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).     

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive

Let K be a field of characteristics 0 complete with respect to a discrete valuation v, with a perfect residue field of characteristic p>0. Let be an algebraic closure of K and K_nr its maximal unramified subextension. Let E be an elliptic curve over K with an integral modular invariant. The curve E has potentially good reduction at v, and there exists a smallest extension L of K_nr over which E has good reduction at v. The Galois group, Gal (L/K_nr) is known in the case p≥5. In this paper we give receipts to determine this group in the cases p=2 and p=3.      

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.      

Automorphismen Von Cayleyalgebren Über Algebraischen Erweiterungen Endlicher KÖrper Ungerader Charakteristik

Let C denote the (split) Cayley algebra over a finite field K of odd characteristic. Given any automorphism σ of C, which is not expressible as the product of two involutory automorphisms, we show that the minimal polynomial of σ is (x ? l)(x² + x + 1)³]. This result remains true, if K is replaced by an infinite algebraic extension K′ of K. Furthermore the automorphism group of C over K′ is bireflectional iff every polynomial of degree 3 in K′[x] is reducible. This corrects and extends the results achieved by Huberta Lausch in [2].     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Potentially good reduction of Barsotti-Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K^′ of K. We prove that there exists a constant c?2 which depends on the absolute ramification index e(K^′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron-Ogg-Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge-Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

A Remark on an Algebraic Riemann—Roch Formula for Flat Bundles

Let f:X S be a smooth projective morphism over an algebraically closed field, with X and S regular. When E, ) is a flat bundle over X, then its Gauss–Manin bundles on S have a flat connection and one may ask for a Riemann–Roch formula relating the algebraic Chern–Simons and Cheeger–Simons invariants. We give an answer for X = Y × S, f = projection. The method of proof is inspired by the work of Hitchin and Simpson.     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.     

Garlands containing the general linear group over an intermediate field

Let K/k be a finite extension of fields with an intermediate subfield L, and let H = GL_L(K) be the general linear group of all L-linear invertible mappings of the vector space of the field K over L. It is proved that the subgroups lying between GL_K(K)^H and the normalizer of H in G, where G = GL_k(K), form a garland. Bibliography: 4 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 34–41.     

Modular Counting of Rational Points over Finite Fields

Let F_q be the finite field of q elements, where q = p^h. Let f(x) be a polynomial over F_q in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in F_q. In this paper we give a deterministic algorithm which computes the reduction of N(f) modulo p^b in O(n(8m)p^(h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed by Gopalan, Guruswami, and Lipton.     

Weak approximation over function fields of curves over large or finite fields

Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with ${X(K)\neq\emptyset}Let K = k(C) be the function field of a curve over a field k and let X be a smooth, projective, separably rationally connected K-variety with X(K) 1 ?{X(K)\neq\emptyset}. Under the assumption that X admits a smooth projective model p: X? C{\pi: \mathcal{X}\to C}, we prove the following weak approximation results: (1) if k is a large field, then X(K) is Zariski dense; (2) if k is an infinite algebraic extension of a finite field, then X satisfies weak approximation at places of good reduction; (3) if k is a nonarchimedean local field and R-equivalence is trivial on one of the fibers X_p{\mathcal{X}_p} over points of good reduction, then there is a Zariski dense subset W í C(k){W\subseteq C(k)} such that X satisfies weak approximation at places in W. As applications of the methods, we also obtain the following results over a finite field k: (4) if |k| > 10, then for a smooth cubic hypersurface X/K, the specialization map X(K)? ?_{p ? P}X_p(k(p)){X(K)\longrightarrow \prod_{p\in P}\mathcal{X}_p(\kappa(p))} at finitely many points of good reduction is surjective; (5) if char k 1 2, 3{\mathrm{char}\,k\neq 2,\,3} and |k| > 47, then a smooth cubic surface X over K satisfies weak approximation at any given place of good reduction.     

Hilbertianity of division fields of commutative algebraic groups

Let G be a commutative algebraic group over a finitely generated infinite field K of characteristic p. We prove that every extension of K contained in the field obtained by adjoining to K all prime-to-p torsion points of G is Hilbertian. We also determine when the field obtained by adjoining to K all torsion points of G has this property. This extends results of Moshe Jarden on abelian varieties.