Related papers: On Weil-Stark elements, II: refined Stark conjectu…
We develop a detailed arithmetic theory related to special values at arbitrary integers of the Artin $L$-series of linear characters. To do so we define canonical generalized Stark elements of arbitrary `rank' and `weight', thereby…
We prove a conjecture on Rubin-Stark elements, which was recently proposed by the author, and also by Mazur and Rubin, in a special case.
A theory of cyclic elements in semisimple Lie algebras is developed. It is applied to an explicit construction of regular elements in Weyl groups.
We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a $p$-adic version of Rubin-Stark Conjecture and Burns…
The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence…
We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of…
In this paper, we present a unifying approach to the general theory of abelian Stark conjectures. To do so we define natural notions of `zeta element', of `Weil-\'etale cohomology complexes' and of `integral Selmer groups' for the…
We formulate a generalization of a `refined class number formula' of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the `order of vanishing' and the `leading term'.…
In this paper, we give a form of refined Roth's theorem. As an application, we prove a special case of the $abc$-conjecture.
First a few reformulations of Frankl's conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to…
In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.
We construct a canonical family of elements in the reduced exterior power lattices of the unit groups of global fields. We prove that this family recovers the theory of cyclotomic elements in real abelian fields and also establish detailed…
Dual presentations of Coxeter groups have recently led to breakthroughs in our understanding of affine Artin groups. In particular, they led to the proof of the $K(\pi, 1)$ conjecture and to the solution of the word problem. Will the "dual…
We propose a relatively new notion of two-valued elements, which arises naturally in constructing the star exponential functions of the quad-ratics in the Weyl algebra over the complex number field. This notion enables us to describe the…
We present some new results on the cohomology of a large scope of SL\_2-groups in degrees above the virtual cohomological dimension; yielding some partial positive results for the Quillen conjecture in rank one. We combine these results…
We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of…
In a previous paper the second author developed a new approach to the abelian p-adic Stark Conjecture at s=1 and stated some related conjectures. This paper develops and applies techniques using p-adic measures and continued fractions to…
We introduce a new version $kk^{\rm alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the…
We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except…
Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…