Multiplicities of mod p Galois representations

We study the multiplicity with which 2-dimensional mod p Galois representations occur in Jacobians of modular curves. Received: 26 February 1997 / Revised version: 2 July 1997     

Pro-p Galois groups of rank ≤ 4

Let p be a prime > 2, let F be a field of characteristic ≠p containing a primitive p-th root of unity and let G _F (p) be the Galois group of the maximal Galois-p-extension of F. If rk G _F (p)≤ 4 then G _F (p) is a free pro-p product of metabelian groups or G _F (p) is a Demuškin group of rank 4. Received: 3 September 1997 / Revised version: 3 October 1997     

Small maximal pro-p Galois groups

For an odd prime p we classify the pro-p groups of rank ≤ 4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of order p. Received: 2 September 1997     

On totally monotonic families of sequences

Summary. Total positivity is an important concept which has been studied by Gantmacher, Karlin and Schoenberg. In section 1, new properties of totally positive functions are given. The concept of totally monotonic families of sequences (TMFS), by mean of positivity of some determinant, is defined and studied. Applications to extrapolation algorithms are given. Received October 22, 1997 / Revised version received April 14, 1998 / Published online August 19, 1999     

Geometry of real polynomial mappings

In this paper we study the set of points at which a real polynomial mapping is not proper. Received: 7 May 1999; in final form: 18 September 2000 / Published online: 25 June 2001     

Holomorphic hulls in terms of curves of nonnegative divisors and generalized polynomial hulls

This paper is devoted to the study of the Basner hulls of compact subsets of ℂ ^N . The principal result is a characterization of these hulls in terms of continuous families of varieties analogous to a well known characterization of polynomially convex hulls. This result is based on recent work of the authors concerning continuous families of divisors. Received: 30 March 1998 / Revised version: 27 September 1998     

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over , and X a compact connected Riemann surface. Let be a Levi factor of some parabolic subgroup of G, with its maximal abelian quotient. We prove that a holomorphic G-bundle over X admits a flat connection if and only if for every such L and every reduction of the structure group of to L, the -bundle obtained by extending the structure group of is topologically trivial. For , this is a well-known result of A. Weil. Received: 1 December 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

Pluripolarity of sets with small Hausdorff measure

We show that any set E⊂C ⁿ , n≥ 2, with finite Hausdorff measure? is pluripolar. The result is sharp with respect to the measuring function. The new idea in the proof is to combine a construction from potential theory, related to the real variational integral , , with properties of the pluricomplex relative extremal function for the Bedford–Taylor capacity. Received: 20 May 1999     

Arithmetic families of smooth surfaces and equisingularity of embedded schemes

This article deals with the foundations of a theory of equisingularity for families of zero-dimensional sheaves of ideals on smooth algebraic surfaces, in the arithmetic context, i.e., where one works with schemes defined over Dedekind rings. Here, different equisingularity conditions are analyzed and compared, based on one of the following requirements: 1) each member of the the family has the same desingularization tree, 2) the family admits a simultaneous desingularization, 3) a naturally associated family of curves is equisingular. Similar conditions had been investigated, in the context of Complex Local Analytic Geometry, by J. J. Risler. Received: 17 November 1997 / Revised version: 19 April 1999     

On Macaulayfication of sheaves

Let X be a quasi-projective scheme and ℱ a coherent sheaf of modules over X such that its non-Cohen–Macaulay locus is at most one dimensional. We use and extend the techniques of Brodmann to construct proper birational morphisms of quasi-projective schemes f:Y→X and Cohen–Macaulay coherent sheaves of modules over Y that are isomorphic to the pull-back of ℱ away from the exceptional locus of f. Certain blow-ups of X at locally complete intersections subschemes which contain non-reduced scheme structures on the non-Cohen–Macaulay locus of ℱ are the main part of the construction. Received: 19 February 1998 / Revised version: 28 December 1998     

Explicit valuations of division polynomials¶of an elliptic curve

In this paper, we estimate valuations of division polynomials and compute them explicitely at singular primes. We show that ν_?(ψ _m (M)) is asymptotically equal to ν_?(m) for a non-torsion point M such that M mod ? is non-zero and non-singular, and it is asymptotically equal to c ₁ m ¹ for some constant c ₁ for a non-torsion point M such that M mod ? is either singular or zero. Furthermore, we show that the common factors of φ _m (M) and ψ _m ²(M) have valuations at ? asymptotically equal to c ₂ m ² for some constant c ₂ when M mod ? is singular, which is a generalization of M. Ayad's result. Received: 10 July 1997 / Revised version: 11 May 1998