Lifting property for finite groups
We classify all finite groups that have lifting property of mod $p$ representations to mod $p^2$ representations for all prime $p$.
math.GR↗
arXiv subjects
Alexander Merkurjev
Publications and source records attributed to Alexander Merkurjev.
The lifting problem for Galois representations
We solve the lifting problem for Galois representations in every dimension and in every characteristic. That is, we determine all pairs $(n,k)$, where $n$ is a positive integer and $k$ is a field of characteristic $p>0$, such that for every field $F$, every continuous homomorphism $\Gamma_F\to \mathrm{GL}_n(k)$ lifts to $\mathrm{GL}_n(W_2(k))$, where $\Gamma_F$ is the absolute Galois group of $F$ and $W_2(k)$ is the ring of $p$-typical length $2$ Witt vectors of $k$.
math.NT↗
Galois representations modulo $p$ that do not lift modulo $p^2$
For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime $p$, every integer $n\geq 3$, and every field $F$ containing a primitive $p$-th root of unity, there exists a continuous $n$-dimensional mod $p$ representation of the absolute Galois group of $F(x_1,\dots,x_p)$ which does not lift modulo $p^2$. This answers a question of Khare and Serre, and disproves a conjecture of Florence.
math.NT↗
Degenerate fourfold Massey products over arbitrary fields
We prove that, for all fields $F$ of characteristic different from $2$ and all $a,b,c\in F^\times$, the mod $2$ Massey product $\langle a,b,c,a \rangle$ vanishes as soon as it is defined. For every field $F_0$, we construct a field $F$ containing $F_0$ and $a,b,c,d\in F^\times$ such that $\langle a,b,c \rangle$ and $\langle b,c,d \rangle$ vanish but $\langle a,b,c,d \rangle$ is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod $2$ cochain DGA of the absolute Galois group is not formal.
math.NT↗
Non-formality of Galois cohomology modulo all primes
Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$. We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times}$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm{Br}(F)[p]$ but $\langle a,b,c,d\rangle$ is not defined over $L$. Thus the Strong Massey Vanishing Conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^*(\Gamma_L,\mathbb{Z}/p\mathbb{Z})$ of the absolute Galois group $\Gamma_L$ of $L$ is not formal. This answers a question of Positselski.
math.NT↗
On the Massey Vanishing Conjecture and Formal Hilbert 90
Let $p$ be a prime number, let $G$ be a profinite group, let $θ\colon G\to \mathbb{Z}_p^{\times}$ be a continuous character, and for all $n\geq 1$ write $\mathbb{Z}/p^n\mathbb{Z}(1)$ for the twist of $\mathbb{Z}/p^n\mathbb{Z}$ by the $G$-action. Suppose that $(G,θ)$ satisfies a formal version of Hilbert's Theorem 90: for all open subgroups $H\subset G$ and every $n\geq 1$, the map $H^1(H,\mathbb{Z}/p^n\mathbb{Z}(1))\to H^1(H,\mathbb{Z}/p\mathbb{Z}(1))$ is surjective. We show that the Massey Vanishing Conjecture for triple Massey products and some degenerate fourfold Massey products holds for $G$. A key step in our proof is the construction of a Hilbert 90 module for $(G,θ)$: a discrete $G$-module $M$ which plays the role of the Galois module $F_{\text{sep}}^\times$ for the absolute Galois group of a field $F$ of characteristic different from $p$.
math.NT↗
The Massey Vanishing Conjecture for fourfold Massey products modulo 2
We prove the Massey Vanishing Conjecture for $n=4$ and $p=2$. That is, we show that for all fields $F$, if a fourfold Massey product modulo $2$ is defined over $F$, then it vanishes over $F$.
math.NT↗
Operations in connective K-theory
In this article we classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the $\hat{\mathbb{Z}}$ one. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations and relate it to a basis of stable operations in graded K-theory. We classify multiplicative operations in both theories and show that homogeneous additive stable operations with $\hat{\mathbb{Z}}$-coefficients are topologically generated by stable multiplicative operations. This is not true for integral operations.
math.KT↗
The norm principle for type $D_n$ groups over complete discretely valued fields
Let $K$ be a complete discretely valued field with residue field $k$ with $\mathrm{char}(k)\neq 2$. Assuming that the norm principle holds for extended Clifford groups $Ω(q)$ for every even dimensional non-degenerate quadratic form $q$ defined over any finite extension of $k$, we show that it holds for extended Clifford groups $Ω(Q)$ for every even dimensional non-degenerate quadratic form $Q$ defined over $K$.
math.GR↗
On polynomially integrable convex bodies
An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky.
math.MG↗
Invariants of degree 3 and torsion in the Chow group of a versal flag
We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.
math.AG↗