arXiv subjects

Avner Ash

Publications and source records attributed to Avner Ash.

23 records · Page 2Linked to original sources

arXiv2002

Moment Problems and the Causal Set Approach to Quantum Gravity

We study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity. The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition. The corresponding dynamics are completely determined by a sequence of nonnegative real coupling constants. Using techniques related to the classical moment problem, we give a complete description of any such sequence of coupling constants. We prove a representation theorem: every discrete theory of quantum gravity arising from causal set dynamics satisfying covariance, causality and renormalizability corresponds to a unique probability distribution function on the nonnegative real numbers, with the coupling constants defining the theory given by the moments of the distribution.

gr-qc↗

arXiv2001

Galois representations with conjectural connections to arithmetic cohomology

In this paper we extend a conjecture of Ash and Sinnott relating niveau one Galois representation to the mod p cohomology of congruence subgroups of SL(n,Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n=3 in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations which are in no obvious way related to lower dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

math.NT↗

arXiv2000

Cohomology of congruence subgroups of SL(4,Z)

Let $N>1$ be an integer, and let $Γ= Γ_0 (N) \subset \SL_4 (\Z)$ be the subgroup of matrices with bottom row congruent to $(0,0,0,*)\mod N$. We compute $H^5 (Γ; \C) $ for a range of $N$, and compute the action of some Hecke operators on many of these groups. We relate the classes we find to classes coming from the boundary of the Borel-Serre compactification, to Eisenstein series, and to classical holomorphic modular forms of weights 2 and 4.

math.NT↗

arXiv1999

An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z)

The conjecture of Serre referred in the title is the one about modularity of odd Galois representations into GL(2,F) where F is a finite field of characteristic p. We present an analogous conjecture where GL(2) is replaced by GL(n). We explain the analogue of "oddness." We then present some theoretical and experimental evidence for the conjecture, primarily when n = 3. Our conjecture does not require the Galois representation to be irreducible. Our most interesting examples involve the sum of an irreducible even 2-dimensional representation and a 1-dimensional representation.

math.NT↗

arXiv1996

Cohomology at infinity and the well-rounded retract for general Linear Groups

Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let $Γ$ be an arithmetic subgroup of $\bold G(\Q)$. Let $X$ be the symmetric space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion) of the space $X/Γ$ is the same as the cohomology of $Γ$. In turn, $X/Γ$ will have the same cohomology as $W/Γ$, if $W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation retract of $X$ by a $Γ$-equivariant deformation retraction, that $W/Γ$ is compact, and that $\dim W$ equals the virtual cohomological dimension (vcd) of $Γ$. Then $W$ can be given the structure of a cell complex on which $Γ$ acts cellularly, and the cohomology of $W/Γ$ can be found combinatorially.

math.RT↗