Related papers: A generalization of the Ross symbols in higher K-g…

The Ross symbol is defined to be an element {1-z,1-w\} in K_2 of a Fermat curve z^n+w^m=1. Ross showed that it is non-torsion by computing the Beilinson regulator. In this paper, we introduce a generalization of the Ross symbols in K_{d+1}…

We introduce new p-adic convergent functions, which we call the p-adic hypergeometric functions of logarithmic type. The first main result is to prove the congruence relations that are similar to Dwork's. The second main result is that the…

We discuss Beilinson's regulator on K_2 of certain fibrations of algebraic varieties which we call the hypergeomtric fibrations. The main result is to describe regulators via the hypergeometric functions 3F2 or 4F3. We also discuss the…

We define an analogue of the `Real' Deligne cohomology group at a prime of semi-stable or good reduction of a variety. We also define regulator maps to this group and formulate a conjecture about the image. This allows us to formulate a…

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also…

In this paper we study the group K_{2n}^{(n+1)}(F) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson--Soul\'e conjecture on weights. In particular, we compute…

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and…

We derive a power series formula for the $p$-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of $p$. In…

We construct families of hyperelliptic curves over Q of arbitrary genus g with (at least) g integral elements in K_2. We also verify the Beilinson conjectures about K_2 numerically for several curves with g=2, 3, 4 and 5. The paper is…

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-functions for non-critical slope…

We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the…

We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. We construct new…

For a prime $p$, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the…

Restricting ourselves to elliptic curves over $\mathbb{Q}$, we reformulate the $p$-adic Beilinson conjecture due to Perrin-Riou, which is customized to our computational approach. We then develop a new algorithm for numerical verifications…

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of…

We propose a non-commutative generalization of Beilinson's Conjecture on the regulator map from algebraic K-theory to Deligne cohomology of algebraic varieties over Q. We also check a baby case of the generalized conjecture, namely, the…

Using the properties of generalized Fibonacci numbers, we determine the automorphism groups of some K3 surfaces with Picard number 2. Conversely, using the automorphisms of K3 surfaces with Picard number 2, we prove the criterion for a…

We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soule, by…

We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The…

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are…