arXiv subjects
Publications and source records attributed to Tobias Finis.
We prove a general convergence result for zeta functions of prehomogeneous vector spaces extending results of H. Saito, F. Sato and Yukie. Our analysis points to certain subspaces which yield boundary terms. We study it further in the setup arising from nilpotent orbits. In certain cases we determine the residue at the rightmost pole of the zeta function.
In this paper, we study the asymptotic behavior of the traces of Hecke operators for spherical discrete automorphic representations of fixed level on general split reductive groups over $\mathbb{Q}$. Under a condition on the analytic behavior of intertwining operators, which is known for the classical groups and the exceptional group $G_2$, we obtain the expected asymptotics in terms of the spherical Plancherel measure and an explicit estimate for the remainder.
Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.
In this paper, a zeta integral for the space of binary cubic forms is associated with the subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group $G_2$.
In this paper we continue to study the degrees of matrix coefficients of intertwining operators associated to reductive groups over $p$-adic local fields. Together with previous analysis of global normalizing factors we can control the analytic properties of global intertwining operators for a large class of reductive groups over number fields, in particular for inner forms of $GL(n)$ and $SL(n)$ and quasi-split classical groups. This has a direct application to the limit multiplicity problem for these groups.
The motivating question of this paper is roughly the following: given a group scheme $G$ over $\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\mathbb{Q}_p}$, how far are open subgroups of $G(\mathbb{Z}_p)$ from subgroups of the form $X(\mathbb{Z}_p)\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{\mathbb{Q}_p}$ simply connected there exist constants $J\ge1$ and $\varepsilon>0$, depending only on $G$, such that any open subgroup of $G (\mathbb{Z}_p)$ of level $p^n$ admits an open subgroup of index $\le J$ which is contained in $X(\mathbb{Z}_p)\mathbf{K}_p(p^{\lceil \varepsilon n\rceil})$ for some proper connected algebraic subgroup $X$ of $G$ defined over $\mathbb{Q}_p$. Moreover, if $G$ is defined over $\mathbb{Z}$, then $\varepsilon$ and $J$ can be taken independently of $p$. We also give a correspondence between natural classes of $\mathbb{Z}_p$-Lie subalgebras of $\mathfrak{g}_{\mathbb{Z}_p}$ and of closed subgroups of $G(\mathbb{Z}_p)$ that can be regarded as a variant over $\mathbb{Z}_p$ of Nori's results on the structure of finite subgroups of $\operatorname{GL}(N_0,\mathbb{F}_p)$ for large $p$. As an application we give a bound for the volume of the intersection of a conjugacy class in the group $G (\hat{\mathbb{Z}}) = \prod_p G (\mathbb{Z}_p)$, for $G$ defined over $\mathbb{Z}$, with an arbitrary open subgroup. In a future paper, this result will be applied to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice $G (\mathbb{Z})$.
We extend the geometric side of Arthur's non-invariant trace formula for a reductive group $G$ defined over $\mathbb{Q}$ continuously to a natural space $\mathcal{C}(G(\mathbb{A}^1))$ of test functions which are not necessarily compactly supported. The analogous result for the spectral side was obtained in [MR2811597]. The geometric side is decomposed according to the following equivalence relation on $G(\mathbb{Q})$: $γ_1\simγ_2$ if $γ_1$ and $γ_2$ are conjugate in $G(\bar{\mathbb{Q}})$ and their semisimple parts are conjugate in $G(\mathbb{Q})$. All terms in the resulting decomposition are continuous linear forms on the space $\mathcal{C}(G(\mathbb{A})^1)$, and can be approximated (with continuous error terms) by naively truncated integrals.
We provide a uniform estimate for the $L^1$-norm (over any interval of bounded length) of the logarithmic derivatives of global normalizing factors associated to intertwining operators for the following reductive groups over number fields: inner forms of $GL(n)$; quasi-split classical groups and their similitude groups; the exceptional group $G_2$. This estimate is a key ingredient in the analysis of the spectral side of Arthur's trace formula. In particular, it is applicable to the limit multiplicity problem studied by the authors in earlier papers.
This is a sequel to arXiv:1308.3604. We study applications to limit multiplicity generalizing the results of arXiv:1208.2257.
We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups GL(n) and SL(n) we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur's trace formula obtained in [FLM11, FL11], which allows us to show that for GL(n) and SL(n) the contribution of the continuous spectrum is negligible in the limit.
We study the exactness of certain combinatorially defined complexes which generalize the Orlik-Solomon algebra of a geometric lattice. The main results pertain to complex reflection arrangements and their restrictions. In particular, we consider the corresponding relation complexes and give a simple proof of the $n$-formality of these hyperplane arrangements. As an application, we are able to bound the Castelnouvo-Mumford regularity of certain modules over polynomial rings associated to Coxeter arrangements (real reflection arrangements) and their restrictions. The modules in question are defined using the relation complex of the Coxeter arrangement and fiber polytopes of the dual Coxeter zonotope. They generalize the algebra of piecewise polynomial functions on the original arrangement.
This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2,O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n^2 / log n) for any fixed lattice Gamma in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.