We prove a version of the weight part Serre's conjecture for mod $p$ Galois representations attached to automorphic forms on rank 2 unitary groups which are non-split at $p$. More precisely, let $F/F^+$ denote CM extension totally real field such that every place $F^+$ above is unramified and inert in $F$, $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+) \longrightarrow {}^C\mathbf{U}_2(\overlin...

Let F be the function field of a curve over p-adic field. D/F central division algebra prime exponent $\ell$ which is different from p. Assume that contains primitive ${\ell}^2$-th root unity. Then group $SK_1(D):=SL_1(D)/[D^*, D^*]$ trivial.

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the BlochKato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal...

The mathematical basis of p-adic Higgs mechanism discussed in papers [email protected] 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical identification between positive real numbers and p-adic numbers are described. Canonical identification induces p-adic topology and differentiable structure on real axis ...

Let p be a prime number, K a number field, and S a finite set of places of K. Let KS be the compositum of all extensions of K (in a fixed algebraic closure K) which are unramified outside S, and put GK,S = Gal(KS/K) for its Galois group. These arithmetic fundamental groups play a very important role in number theory. Algebraic geometry provides the most fruitful known source of information conc...