arXiv subjects
Publications and source records attributed to Chandrasheel Bhagwat.
In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois representations.
Let $Γ_1$ and $Γ_2$ be two lattices of finite covolume in a semisimple Lie group $G$. We prove a spectral rigidity result for the representation spectra of the right regular representations $L^2(Γ_1 \backslash G)$ and $L^2(Γ_2 \backslash G)$ of $G$. This can be thought of as an analogue of the strong multiplicity one theorem and it generalises a result by the first author and Rajan to the case of non-uniform lattices.
This article is inspired from the work of M Krithika and P Vanchinathan on Cluster Magnification and the work of Alexander Perlis on Cluster Size. We establish the existence of polynomials for given degree and cluster size over number fields which generalises a result of Perlis. We state the Strong cluster magnification problem and establish an equivalent criterion for that. We also discuss the notion of weak cluster magnification and prove some properties. We provide an important example answering a question about Cluster Towers. We introduce the concept of Root capacity and prove some of its properties. We also introduce the concept of unique descending and ascending chains for extensions and establish some properties and explicitly compute some interesting examples. We establish results about all these phenomena under a particular type of base change and discuss some other related results about strong cluster magnification and unique chains. The article concludes with results about ascending index for a field extension which are analogous to results about cluster size.
We prove an analogue of the strong multiplicity one theorem in the context of $τ_n$-spherical representations of the group $G = SO(2,1)^\circ$ appearing in $L^2(Γ_i \backslash G)$ for uniform torsion-free lattices $Γ_i, i = 1, 2$ in $G$. This is a generalisation of a previous result by the first author and C. S. Rajan in \cite{B-R-2011} for the case of $G = SO(2,1)^\circ$.
Let $(τ, V_τ)$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $Γ$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $τ$ and multiplicities of irreducible $τ^\vee$-spherical spectra in $L^2(Γ\backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $τ$-isospectrality. Along with this we prove that the almost equality of $τ$-spherical spectra of two lattices assures the equality of their $τ$-spherical spectra.
For an even positive integer $n$, we study rank-one Eisenstein cohomology of the split orthogonal group ${\rm O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands $L$-functions associated to the group ${\rm GL}_1 \times {\rm O}(2n)$ over $F$. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin - Selberg $L$-functions attached to a pair of Hilbert modular forms.
In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric cube transfer of automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of a given weight and with varying level structure. This generalises the recent work of C. Ambi [2020] about the similar problem for ${\rm GL}_3$.
In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\rm GL}_2$ and the automorphic tensor product of two representations of ${\rm GL}_2$ can not be equal (up to a twist by a character of ${\rm GL}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.
One can define the notion of primitive length spectrum for a simple regular periodic graph via counting the orbits of closed reduced primitive cycles under an action of a discrete group of automorphisms. We prove that this primitive length spectrum satisfies an analogue of the `Multiplicity one' property. We show that if all but finitely many primitive cycles in two simple regular periodic graphs have equal lengths, then all the primitive cycles have equal lengths. This is a graph-theoretic analogue of a similar theorem in the context of geodesics on hyperbolic spaces. We also prove, in the context of actions of finitely generated abelian groups on a graph, that if the adjacency operators for two actions of such a group on a graph are similar, then corresponding periodic graphs are length isospectral.
In this paper we establish a strong multiplicity one type property for the length-holonomy spectrum for the three dimensional compact hyperbolic spaces. We use the analytic properties of Selberg-Gangolli-Wakayama zeta functions associated to compact hyperbolic spaces.
This is an announcement of certain rationality results for the critical values of the degree-2n L-functions attached to GL(1) $\times$ SO(n, n) over $\mathbb Q$ for an even positive integer n. The proof follows from studying the rank-one Eisenstein cohomology for SO(n + 1, n + 1).
Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbers in the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growing complex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free but has a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering, which we call "stretching similarity", in this network. We also show that the various characteristics like average degree, global clustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analytical arguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numerical analysis.
In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed by Gopal Prasad and A. S. Rapinchuk where instead of representation equivalence, the lattices under consideration are weakly commensurable Zariski dense subgroups.
In this article we study the nonvanishing of cuspidal cohomology for GL(n). Using endoscopic transfer from various classical groups we construct cuspidal representations of GL(n) of cohomological type while working over a totally real field or a totally imaginary quadratic extension of a totally real field. Generalizing a construction of Laurent Clozel, we also prove nonvanishing of cuspidal cohomology of GL(2n) over any number field but only for coefficient systems coming from parallel weights. Working at an arithmetic level, we also draw some inferences on an endoscopic stratification of inner cohomology of GL(n).
In this paper, we give a description of Deligne's periods $c^\pm$ for tensor product of pure motives $M \otimes M'$ over $\mathbb{Q}$ in terms of the period invariants attached to $M$ and $M'$ by Yoshida. The period relations proved by the author and Raghuram in an earlier paper follow from the results of this paper.
In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for ${\rm GL}_n \times {\rm GL}_{n'}$ proved by Günter Harder and the second author.
Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric spaces isospectral with respect to the Laplacian on functions. For this, in higher ranks, they assume the validity of Schanuel's conjecture. In this note, we observe that if we use the stronger notion of representation equivalence of lattices, then Schanuel's conjecture can be avoided. Further, the results are also applicable in a $S$-arithmetic setting. We also introduce a new relation on the class of arithmetic lattices, stronger than weak commensurability, which we call as characteristic equivalence, and show that it simplifies some of the arguments used in Prasad and Rapinchuk (2009) to deduce commensurability type results from weak commensurability.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.