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     

Cohomology of tori over p-adic curves

 We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve. Received: 28 January 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 14G20, 14F22, 14L15, 11S25     

Abelian varieties over cyclotomic fields with good reduction everywhere

 For every conductor f{1,3,4,5,7,8,9,11,12,15} there exist non-zero abelian varieties over the cyclotomic field Q(ζ _f ) with good reduction everywhere. Suitable isogeny factors of the Jacobian variety of the modular curve X ₁ (f) are examples of such abelian varieties. In the other direction we show that for all f in the above set there do not exist any non-zero abelian varieties over Q(ζ _f ) with good reduction everywhere except possibly when f=11 or 15. Assuming the Generalized Riemann Hypothesis (GRH) we prove the same result when f=11 and 15. Received: 19 April 2001 / Revised version: 21 October 2001 / Published online: 10 February 2003     

Banach space representations and Iwasawa theory

We develop a duality theory between the continuous representations of a compactp-adic Lie groupG in Banach spaces over a givenp-adic fieldK and certain compact modules over the completed group ringo _K[[G]]. We then introduce a “finiteness” condition for Banach space representations called admissibility. It will be shown that under this duality admissibility corresponds to finite generation over the ringK[[G]]: =K ⊗o _K[[G]]. Since this latter ring is noetherian it follows that the admissible representations ofG form an abelian category. We conclude by analyzing the irreducibility properties of the continuous principal series of the groupG: = GL₂(ℤ _p ).     

On projective curves of maximal regularity

 Let 𝒞⊆ℙ ^r _K be a non-degenerate projective curve of degree d>r+1 of maximal regularity so that 𝒞 has an extremal secant line . We show that 𝒞∪ is arithmetically Cohen Macaulay if d<2r−1 and we study the Betti numbers and the Hartshorne-Rao module of the curve 𝒞. Received: 27 March 2002; in final form: 24 May 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 14H45, 13D02. The second author was partially supported by Swiss National Science Foundation (Projects No. 20-52762.97 and 20-59237.99).     

A relative class number formula for an imaginary abelian number field by means of Dedekind sum

 We give a relative class number formula for an imaginary abelian number field by means of some ``Dedekind sum'. This is a generalization of Carlitz and Olson's formula presented in 1955. Received: 18 December 2001 / Revised version: 15 May 2002 Mathematics Subject Classification (2000): 11R29, 11R20     

On unramified Galois extensions constructed using Galois representations

 For a field k, We denote the maximal abelian extension of k by k _ab and (K _ab ^r−1 _ab by k _ab ^r . In this paper, unramified Galois extensions over k _ab ^r are constructed using Galois representations of arbitrary dimension with larger coefficient rings. Received: 31 August 2001 / Revised version: 22 March 2002 Mathematics Subject Classification (2000): 11R21     

Torsion Completeness of Sylow <Emphasis Type="Italic">p</Emphasis>-Groups in Semisimple Group Rings

Let G be an abelian p-group and K be a field of the first kind with respect to p of char K ≠p and of sp(K) = N or NU {0}. Then it is shown that the normed Sylow p-subgroup S(KG) is torsion complete if and only if G is bounded (Theorem 1). An analogous fact is proved for the case when K is of the second kind (Theorem 2). These completely settle a conjecture posed by us in Compt. Rend. Acad. Bulg. Sci. (1993) and are also a supplement to our result in the modular case published in Acta Math. Hungar. (1997).     

Équivalence rationnelle sur les hypersurfaces cubiques sur les corps <Emphasis Type="Italic">p</Emphasis>-adiques

 We study R-equivalence on cubic hypersurfaces, and explain how to construct families of rational curves. We show that for a smooth cubic hypersurface defined over a number field K, the Chow group of zero-cycles of degree 0 is trivial in almost every place of K. Received: 20 March 2002 Published online: 24 January 2003     

On Abelian Branched Coverings of the Sphere

 We obtain an enumeration formula for the number of weak equivalence classes of the branched (?×ℬ)-covering of the sphere with m-branch points, when ? and ℬ are finite abelian groups with (|?|,|ℬ|)=1. From this, we can deduce an explicit formula for enumerating the weak equivalence classes of pseudofree spherical (ℤ _p ×ℤ _q )-actions on a given surface, when p and q are distinct primes. Received: August 10, 1999 Final version received: June 19, 2000     

Some stable vector bundles with reducible theta divisor

 Let C be a curve of genus g and L a line bundle of degree 2g on C. Let M_L be the kernel of the evaluation map . We show that when L is general enough, the rank g bundle M_L and its exterior powers are stable, but admit a reducible theta divisor. Received: 16 September 2002 / Revised version: 29 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 14H60     

Universal abelian covers of quotient-cusps

 The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity. Received: 3 February 2001 / Revised version: 8 March 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 14B05, 14J17, 32S25 This research was supported by grants from the Australian Research Council and the NSF (first author) and the the NSA (second author).     

Stein extensions of real symmetric spaces and the geometry of the flag manifold

 Let G be a connected semisimple Lie group contained in its simply connected complexification G _C . Let KG∩K _C be a maximal compact subgroup of G. Denote by X _o the unique closed G-orbit in the full flag manifold ℱ and by 𝒪 the unique open K _C -orbit in ℱ. The set consisting of the elements gK _C so that gX _o ⊂𝒪 is an Stein extension of G/K⊂G _C /K _C . There is a universal domain , natural form the point of view of group actions which has been conjectured to be Stein. The main result of this paper is the inclusion . In the second part of the paper I show, under some dominance condition in the parameter, that representations in Dolbeault cohomology can be realized as holomorphic sections of vector bundles over . Received: 9 September 2002 / Revised version: 12 July 2002 / Published online: 8 April 2003 Mathematics Subject Classification (2002): 22E30 Research partially supported by NSF grant DMS-9801605 and DMS 0074991.     

Toward equivariant Iwasawa theory

 Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G _∞ , X _∞ the Galois groups of K _∞ /k and M _∞ /K _∞ , respectively, where K _∞ is the cyclotomic l-extension of K and M _∞ the maximal abelian S-ramified l-extension of K _∞ with S a sufficiently large finite set of primes of k. We introduce a new K-theoretic variant of the Iwasawa ℤ[[G _∞ ]]-module X _∞ and, for K/k abelian, formulate a conjecture, which is the main conjecture of classical Iwasawa theory when lł[K : k]. We prove this new conjecture when Iwasawa's μ-invariant vanishes and discuss consequences for the Lifted Root Number Conjecture at l. Received: 7 August 2001 / Revised version: 6 May 2002 We acknowledge financial support provided by NSERC. Mathematics Subject Classification (2000): 11R23, 11R27, 11R32, 11R33, 11R37, 11R42, 11S20, 11S23, 11S31, 11S40     

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ^∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ^∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations. T. Dokchitser is supported by a Royal Society University Research Fellowship.     

Indecomposable cycles on special quartics in ℙ<Superscript>3</Superscript>

 In this note we give an example of an indecomposable higher Chow cycle on a special family of quartics in ℙ³. The example is obtained as an extension of a cycle in the higher Chow group CH ² (K,1) of a singular Kummer surface. Received: 19 June 2002 / Revised version: 17 October 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14C15, 14C25, 14H10, 14H28, 14F42     

On the Malgrange condition for complex polynomials of two variables

 We show that if P  ℂ[X, Y] is nonconstant, the set K _∞ (P) of points where the Malgrange condition fails to hold coincides with the roots of a constructible polynomial in one variable. This strengthens weaker related results in the literature. The proof given here is purely analytical and bypasses the algebraic and topological arguments involved in other approaches. Applications to both the real and complex Jacobian Conjectures and to the reducibility of P-z are discussed. Received: 11 October 2001 / Revised version: 2 April 2002 Mathematics Subject Classification (2000): 12D05, 14D06, 14R15, 32B10     

On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50