We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

This is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal kleinian group.

The Littlewood conjecture in Diophantine approximation claims that$$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·\parallel q\beta \parallel =0$$holds for all real numbers $\alpha$ and $\beta$, where $\parallel ·\parallel$ denotes the distance to the nearest integer. Its $p$-adic analogue, formulated by de Mathan and Teulié in 2004, asserts that$$\underset{q\ge 1}{inf}q·\parallel q\alpha \parallel ·{\left|q\right|}_p=0$$holds for every real number $\alpha$ and every prime number $p$, where ${|·|}_p$ denotes the $p$-adic absolute value normalized by ${\left|p\right|}_p=p^{-1}$. We survey the known results on these conjectures and highlight recent developments.

Let $\Gamma \setminus {ℍ}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \mathrm{SL}(2,ℂ)$ is a discrete subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde{\Gamma }$ is a finite index group extension of $\Gamma$ in $\mathrm{SL}(2,ℂ)$, and $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$, for an appropriate...

Fix an element $\alpha$ in a quadratic field $K$. Define $S$ as the set of rational primes $p$, for which $\alpha$ has maximal order modulo $p$. Under the assumption of the generalized Riemann hypothesis, we show that $S$ has a density. Moreover, we give necessary and sufficient conditions for the density of $S$ to be positive.

Let $k$ be a finite extension of ${\mathbb{Q}}_p$, let $k_1$, respectively $k_3$, be the division fields of level $1$, respectively $3$, arising from a Lubin-Tate formal group over $k$, and let $\Gamma =$ Gal($k_3/k_1$). It is known that the valuation ring $k_3$ cannot be free over its associated order $𝔄$ in $K\Gamma$ unless $k={\mathbb{Q}}_p$. We determine explicitly under the hypothesis that the absolute ramification index of $k$ is sufficiently large.