Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.     

Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map has trivial kernel, denoting the set of places of k.The conjecture is true if G is of type 1A∗, i.e., isomorphic to SL₁(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin [S1] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2A∗, B_n, C_n, D_n (D₄ nontrialitarian), G₂ or F₄; a group is said to be of type 2A∗, if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.     

Operations in A-theory

A construction for Segal operations for K-theory of categories with cofibrations, weak equivalences and a biexact pairing is given and coherence properties of the operations are studied. The model for K-theory, which is used, allows coherence to be studied by means of (symmetric) monoidal functors. In the case of Waldhausen A-theory it is shown how to recover the operations used in Waldhausen (Lecture Notes in Mathematics, Vol. 967, Springer, Berlin, 1982, pp. 390-409) for the A-theory Kahn-Priddy theorem. The total Segal operation for A-theory, which assembles exterior power operations, is shown to carry a natural infinite loop map structure. The basic input is the un-delooped model for K-theory, which has been developed from a construction by Grayson and Gillet for exact categories in Gunnarsson et al. (J. Pure Appl. Algebra 79 (1992) 255), and Grayson's setup for operations in Grayson (K-theory (1989) 247). The relevant material from these sources is recollected followed by observations on equivariant objects and pairings. Grayson's conditions are then translated to the context of categories with cofibrations and weak equivalences. The power operations are shown to be well behaved w.r.t. suspension and are extended to algebraic K-theory of spaces. Staying close with the philosophy of Waldhausen (1982) Waldhausen's maps are found. The Kahn-Priddy theorem follows from splitting the “free part” off the equivariant theory. The treatment of coherence of the total operation in A-theory involves results from Laplaza (Lecture Notes in Mathematics, Vol. 281, Springer, Berlin, 1972, pp. 29-65) and restriction to spherical objects in the source of the operation.     

Algebraic K-theory of special groups

Following the introduction of an algebraic K-theory of special groups in [Dickmann and Miraglia, Algebra Colloq. 10 (2003) 149-176], generalizing Milnor's mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean powers of special groups. A parallel theme is the preservation by these constructions of property [SMC], an analog for the K-theory of special groups of the property “multiplication by l(-1) is injective” in Milnor's mod 2 K-theory (see [Milnor, Invent. Math. 9 (1970) 318-344]).     

Stark conjectures for CM elliptic curves over number fields

We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s=0. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

Ideal lattices over totally real number fields and Euclidean minima

No Abstract. . Received: 14 February 2005     

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that α_i=(a)∪(b_i), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma.     

Distribution of matrices with restricted entries over finite fields

For a prime p, we consider some natural classes of matrices over a finite field F_p of p elements, such as matrices of given rank or with characteristic polynomial having irreducible divisors of prescribed degrees. We demonstrate two different techniques which allow us to show that the number of such matrices in each of these classes and also with components in a given subinterval [-H, H] [-(p - 1)/2, (p - 1)/2] is asymptotically close to the expected value.     

Bounds for the number of points on curves over finite fields

Let U be a bounded open subset of ?^d, d ≥ 2 and f ∈ C(?U). The Dirichlet solution f^CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure ū of U in general if U is not regular but it is always Baire-one.Let H(U) be the space of all functions continuous on the closure ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f^CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f^CU belongs to the subclass B_1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10].A generalization to the abstract context of simplicial function space on a metrizable compact space is provided.We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].     

On K(3)4 of curves over number fields

In this paper we consider the group K ₄ ⁽³⁾ (F) of a field, in case F is the function field of a smooth geometrically irreducible curve over a number field. We do this using the complexes constructed in [4], together with an auxiliary complex. On the image in K ₄ ⁽³⁾ (F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K ₄ ⁽³⁾ (E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson. Oblatum 10-IX-1995 & 8-I-1996     

Indices of hyperelliptic curves over p-adic fields

Let k be a p-adic field of odd residue characteristic and let C be a hyperelliptic (or elliptic) curve defined by the affine equation Y ²=h(X). We discuss the index of C if h(X)=ɛf(X), where ɛ is either a non-square unit or a uniformising element in O _k and f(X) a monic, irreducible polynomial with integral coefficients. If a root θ of f generates an extension k(θ) with ramification index a power of 2, we completely determine the index of C in terms of data associated to θ. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for such curves C. Received: 14 July 1997 / Revised version: 16 February 1998     

Reducibility number

Let P be a poset in a class of posets P. A smallest positive integer r is called reducibility number of P with respect to P if there exists a non-empty subset S of P with |S|=r and P-S∈P. The reducibility numbers for the power set 2ⁿ of an n-set (n?2) with respect to the classes of distributive lattices, modular lattices and Boolean lattices are calculated. Also, it is shown that the reducibility number r of the lattice of all subgroups of a finite group G with respect to the class of all distributive lattices is 1 if and only if the order of G has at most two distinct prime divisors; further if r is a prime number then order of G is divisible by exactly three distinct primes. The class of pseudo-complemented u-posets is shown to be reducible. Deletable elements in semidistributive posets are characterized.     

The number of elliptic curves over ℚ with conductorNwith conductorN

We prove that the number of elliptic curves E/ℚ with conductorN isO(N ^1/2+ε). More generally, we prove that the number of elliptic curves E/ℚ with good reduction outsideS isO(M ^1/2+ε), whereM is the product of the primes inS. Assuming various standard conjectures, we show that this bound can be improved toO(M ^c/loglogM ). Research partially supported by NSF DMS-9424642.     

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P _λ of infinite order in the Mordell-Weil group of the elliptic curve Y ² = X ³ − n ² X over ℚ(λ). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain).     

A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”