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We present an 
$\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.
In this paper we study Zimmer's conjecture for 
$C^{1}$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in 
${\rm SL}(n, {{\mathbb {R}}})$, the dimensional bound is sharp.
In this paper, we prove the assertion that the number of monic cubic polynomials 
$F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over 
${\mathbb {Q}}$ satisfying 
$\max \{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by 
$O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves.
Let 
$\mathbb {V}$ be a motivic variation of Hodge structure on a 
$K$-variety 
$S$, let 
$\mathcal {H}$ be the associated 
$K$-algebraic Hodge bundle, and let 
$\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector 
$v \in \mathcal {H}_{\mathbb {C}, s}$ above 
$s \in S(\mathbb {C})$ which lies inside 
$\mathbb {V}_{s}$, the conjugate vector 
$v_{\sigma } \in \mathcal {H}_{\mathbb {C}, s_{\sigma }}$ is Hodge and lies inside 
$\mathbb {V}_{s_{\sigma }}$. We study this problem in the situation where we have an algebraic subvariety 
$Z \subset S_{\mathbb {C}}$ containing 
$s$ whose algebraic monodromy group 
$\textbf {H}_{Z}$ fixes 
$v$. Using relationships between 
$\textbf {H}_{Z}$ and 
$\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and 
$\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for 
$v$ subject to a group-theoretic condition on 
$\textbf {H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.
For complex simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the commutative subalgebra 
$\mathfrak z(\hat {\mathfrak g})\subset {{\mathcal {U}}}(t^{-1}\mathfrak g[t^{-1}])$ known as the Feigin–Frenkel centre. These formulas make use of the symmetrisation map as well as of some well-chosen symmetric invariants of 
$\mathfrak g$. There are some general results on the rôle of the symmetrisation map in the explicit description of the Feigin–Frenkel centre. Our method reduces questions about elements of 
$\mathfrak z(\hat {\mathfrak g})$ to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G
$_2$ by hand. One of our technical tools is the map 
${\sf m}\!\!: {{\mathcal {S}}}^{k}(\mathfrak g)\to \Lambda ^{2}\mathfrak g \otimes {{\mathcal {S}}}^{k-3}(\mathfrak g)$ introduced here. As the results show, a better understanding of this map will lead to a better understanding of 
$\mathfrak z(\hat {\mathfrak g})$.
${\mathbb {A}}^1$-homotopy theory
For a subring 
$R$ of the rational numbers, we study 
$R$-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in 
${\mathbb {A}}^1$-homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in 
${\mathbb {A}}^1$-homotopy theory, paying attention to future applications for vector bundles. We show that 
$R$-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space 
$BGL_n$ is 
${\mathbb {A}}^1$-nilpotent when 
$n$ is odd, and analyze the (more complicated) situation where 
$n$ is even as well. We establish analogs of various classical results about rationalization in the context of 
${\mathbb {A}}^1$-homotopy theory: if 
$-1$ is a sum of squares in the base field, 
${\mathbb {A}}^n \,{\setminus}\, 0$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres.