Ted Chinburg

Geodesics and commensurability classes of arithmetic hyperbolic

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of QL(M) ∪ QL(M ′) is a rational multiple of the logarithm of a real algebraic number. As noted by Prasad and Rapinchuk in [9], the Gelfond Schneider Theorem [1] implies that a ratio of such logarithms is transcendental if it is irrational. Thus if ` ∈ QL(M)−QL(M ′) then `/`′ is transcendental for all non-zero `′ ∈ QL(M ′). Recently Prasad and Rapinchuk have shown in [9] that if M is an arithmetic hyperbolic manifold of even dimension, then QL(M) and the commensurability class of M determine one another. In addition, they have shown that this is not always true for arithmetic hyperbolic 5-manifolds. However, they have announced a proof that for all locally symmetric spaces associated to a specified absolutely simple Lie group, there are only finitely many commensurability classes of arithmetic lattices giving rise to a given rational length spectrum. It is known (see [4] pp. 415–417) that for closed hyperbolic manifolds, the spectrum of the Laplace-Beltrami operator action on L2(M), counting multiplicities, determines the set of lengths of closed geodesics on M (without counting multiplicities). Hence Theorem 1.1 implies:

3

a class (ON) in the Grothendieck group Ko(Z[G]) of all finitely generated projective Z[G]-modules. The class group Cl(Z[G]) of Z[G] is defined to be the quotient of Ko(Z[G]) by the subgroup generated by the class of Z[G]. Let (ON)stab be the image of (ON) in Cl(Z[G]). In [34] Taylor proved Fr6hlichs conjecture that (ON)stab is equal to another invariant WN/K in Cl(Z[G]) which Cassou-Nogues had defined by means of the root-numbers of symplectic representations of the Galois group G. The root-number of a representation V of G is the constant which appears in the functional equation of the Artin Lfunction of V. The connection between Galois-structure invariants and Artin L-functions has been a basic theme in research on Galois structure; for further discussion, see [12] and [4].

-manifolds

In this chapter we review the construction by Lichtenbaum [8] and Shafare-vitch [11] of relatively minimal and minimal models of curves over Dedekind rings. We have clpsely followed Lichtenbaum [8]; some proofs have been skipped or summarized so as to go into more detail concerning other parts of the construction. Since the main arguments of [8] apply over Dedekind rings, we work always over Dedekind rings rather than discrete valuation rings.

Galois structure of de Rham cohomology of tame covers of schemes

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture.

Minimal Models for Curves over Dedekind Rings

In this chapter we review the basic definitions of Arakelov intersection theory, and then sketch the proofs of some fundamental results of Arakelov, Faltings and Hriljac. Many interesting topics are beyond the scope of this introduction, and may be found in the references [2], [3], [8], [12], [20] and their bibliographies.

Oort groups and lifting problems

We show that the sectional capacity of an adelic subset of a projective variety over a number field is a quasi-canonical limit of arithmetic top self-intersection numbers, and we establish the functorial properties of extremal plurisubharmonic Greens functions. We also present a conjecture that the sectional capacity should be a top self-intersection number in an appropriate adelic arithmetic intersection theory.

An Introduction to Arakelov Intersection Theory

Abstract Suppose Λ is a Brauer tree algebra. We determine the location of a Λ -module M in the stable Auslander–Reiten quiver of Λ from the description of M as a multi-pushout of elementary modules. This is done by introducing a new combinatorial object associated to M , which is a certain walk around the Brauer tree of Λ . These walks have applications to determining universal deformation rings and stable homomorphism groups.

Capacity theory and arithmetic intersection theory

Abstract Let Λ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical J vanishes. We determine the irreducible components of the module variety Rep d ( Λ ) for any dimension vector d. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when Λ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra Λ with J 2 = 0 , our criteria for identifying the components of Rep d ( Λ ) permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on the one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length 2. Finally, we use the transit of information to show that any algebra of Loewy length 2 which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.

Locations of modules for Brauer tree algebras

Let k be a perfect field of characteristic