Showing papers in "Documenta Mathematica in 2005"

On triangulated orbit categories.

Abstract: We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms

Abstract: The L 2 -metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian man- ifold (N;g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic dis- tance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all difieomorphism groups for the L 2 -metric.

Slope filtrations revisited.

Abstract: We give a "second generation" exposition of the slope filtration theorem for modules with Frobenius action over the Robba ring, providing a number of simplifications in the arguments. Some of these are inspired by parallel work of Hartl and Pink, which points out some analogies with the formalism of stable vector bundles.

CM points and quaternion algebras.

Abstract: This paper provides a proof of a technical result (Corol- lary 2.10 of Theorem 2.9) which is an essential ingredient in our proof of Mazur's conjecture over totally real number fields (3).

On the Chow Groups of Quadratic Grassmannians

Abstract: In this text we get a description of the Chow-ring (mod- ulo 2) of the Grassmanian of the middle-dimensional planes on arbi- trary projective quadric. This is only a first step in the computation of the, so-called, generic discrete invariant of quadrics. This generic invariant contains the "splitting pattern" and "motivic decomposi- tion type" invariants as specializations. Our computation gives an important invariant J(Q) of the quadric Q. We formulate a conjecture describing the canonical dimension of Q in terms of J(Q).

On the Equivariant Tamagawa Number Conjecture for Abelian Extensions of a Quadratic Imaginary Field

Abstract: Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extension of k and let K be a subextension of L/k. In this article we prove the p-part of the Equivariant Tamagawa Number Conjecture for the pair (h 0 (Spec(L)), Z(Gal(L/K))).

Arithmetic Characteristic Classes of Automorphic Vector Bundles

Abstract: We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values in an arithmetic Chow ring constructed by means of differential forms with certain log-log type singularities. We first study the cohomolog- ical properties of log-log differential forms, prove a Poincare lemma for them and construct the corresponding arithmetic Chow groups. Then, we introduce the notion of log-singular hermitian vector bun- dles, which is a variant of the good hermitian vector bundles intro- duced by Mumford, and we develop the theory of arithmetic charac- teristic classes. Finally we prove that the hermitian metrics of auto- morphic vector bundles considered by Mumford are not only good but also log-singular. The theory presented here provides the theoretical background which is required in the formulation of the conjectures of Maillot-Roessler in the semi-abelian case and which is needed to extend Kudla's program about arithmetic intersections on Shimura varieties to the non-compact case.

Arakelov invariants of Riemann surfaces

Abstract: We derive closed formulas for the Arakelov-Green func- tion and the Faltings delta-invariant of a compact Riemann surface.

On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves.

Abstract: Let Y0(p) be the Drinfeld modular curve parameterizing Drinfeld modules of rank two over Fq[T ] of general characteristic with Hecke level p-structure, where p ⊳ Fq[T ] is a prime ideal of degree d. Let J0(p) denote the Jacobian of the unique smooth irreducible projective curve containing Y0(p). Define N(p) = q−1 q−1 , if d is odd, and define N(p) = q −1 q2−1 , otherwise. We prove that the torsion subgroup of the group of Fq(T )-valued points of the abelian variety J0(p) is the cuspidal divisor group and has order N(p). Similarly the maximal μ-type finite etale subgroup-scheme of the abelian variety J0(p) is the Shimura group scheme and has order N(p). We reach our results through a study of the Eisenstein ideal E(p) of the Hecke algebra T(p) of the curve Y0(p). Along the way we prove that the completion of the Hecke algebra T(p) at any maximal ideal in the support of E(p) is Gorenstein. 2000 Mathematics Subject Classification: Primary 11G18; Secondary 11G09.

Banach-Hecke algebras and p-adic Galois Representations

Abstract: We take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary Banach space representations of $G(L)$ when $G$ is a split reductive group over $L$. The outline of our work is derived from Breuil's remarkable insights into the nature of the correspondence between 2-dimensional crystalline Galois representations of the Galois group of $\Qdss_p$ and Banach space representations of $GL_{2}(\Qdss_p)$.

A Common Recursion For Laplacians of Matroids and Shifted Simplicial Complexes

Abstract: A recursion due to Kook expresses the Laplacian eigen- values of a matroid M in terms of the eigenvalues of its deletion M −e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M − e, M/e). We further show that by suitably generalizing deletion and contraction to arbitrary simplicial complexes, the Lapla- cian eigenvalues of shifted simplicial complexes satisfy this exact same recursion. We show that the class of simplicial complexes satisfying this recursion is closed under a wide variety of natural operations, and that several specializations of this recursion reduce to basic recursions for natural invariants. We also find a simple formula for the Laplacian eigenvalues of an arbitrary pair of shifted complexes in terms of a kind of generalized degree sequence.

Algebraic K-theory and sums-of-squares formulas

Abstract: We prove a result about the existence of certain 'sums-of- squares' formulas over a field F. A classical theorem uses topological K-theory to show that if such a formula exists over R, then certain powers of 2 must divide certain binomial coefficients. In this paper we use algebraic K-theory to extend the result to all fields not of characteristic 2.

Kida's formula and congruences

Abstract: Let f be a modular eigenform of weight at least two and let F be a finite abelian extension of Q. Fix an odd prime p at which f is ordinary in the sense that the p Fourier coefficient of f is not divisible by p. In Iwasawa theory, one associates two objects to f over the cyclotomic Zp-extension F∞ of F : a Selmer group Sel(F∞, Af ) (where Af denotes the divisible version of the two-dimensional Galois representation attached to f) and a p-adic L-function Lp(F∞, f). In this paper we prove a formula, generalizing work of Kida and Hachimori–Matsuno, relating the Iwasawa invariants of these objects over F with their Iwasawa invariants over p-extensions of F . For Selmer groups our results are significantly more general. Let T be a lattice in a nearly ordinary p-adic Galois representation V ; set A = V/T . When Sel(F∞, A) is a cotorsion Iwasawa module, its Iwasawa μ-invariant μ(F∞, A) is said to vanish if Sel(F∞, A) is cofinitely generated and its λ-invariant λ(F∞, A) is simply its p-adic corank. We prove the following result relating these invariants in a p-extension.

Parametrized braid groups of chevalley groups

Abstract: We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system An; in the second part we prove a similar theorem for the root system Dn. 2000 Mathematics Subject Classification: Primary 20F36; Secondary 19Cxx; 20F55. Keywords and Phrases: Braid group, Steinberg group, parametrized braid group, root system.