The present paper establishes qunatitative estimates on the rate of diophantine approximation in homogeneous varieties of semisimple algebraic groups. The estimates established generalize and improve previous ones, and are sharp in a number of cases. We show that the rate of diophantine approximation is controlled by the spectrum of the automorphic representation, and is thus subject to the generalised Ramanujan conjectures.
Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we establish that the spectrum of the $2n$ exponents of Diophantine approximation in dimension $n\geq3$ is a subset of $\mathbb{R}^{2n}$ with non empty interior.
We introduce the notion of the automorphic dual of a matrix algebraic group defined over $Q$. This is the part of the unitary dual that corresponds to arithmetic spectrum. Basic functorial properties of this set are derived and used both to deduce arithmetic vanishing theorems of ``Ramanujan'' type as well as to give a new construction of automorphic forms.
We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the automorphic fundamental solution of the invariant differential operator (Delta - lambda_z)^N. On one hand, we form a Poincare series from the solution to the corresponding differential equation on the free space G/K, which is obtained using the harmonic analysis of bi-K-invariant functions. On the other hand, a suitable global automorphic Sobolev theory, dev…
The approximation constant $λ_{k}(ζ)$ is defined as the supremum of real $η$ such that $\Vert ζ^{j}x\Vert\leq x^{-η}$ for $1\leq j\leq k$ has infinitely many integer solutions $x$. Here $\Vert.\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $(λ_{1}(ζ),λ_{2}(ζ),\ldots)$ which will lead to various improvements of known results on the individual spectrum of the approximation constants $λ_{k}(ζ)$ as well. In particular, this extends a result by Bugeaud to the case of arbitrary dimension $k$. Concretely, given $k\geq 1$ and $λ\geq 1$, we infer {\e…
Fix an integer $n\ge 2$. To each non-zero point $\mathbf{u}$ in $\mathbb{R}^n$, one attaches several numbers called exponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$ whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called the spectrum of these exponents. We show that, in …
We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find that a generalized truncated Brown-Peterson spectrum BP<2> is an E_\infty ring spectrum at the prime 3.
In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ of form $$ Δ(τ_1, b_1) \boxplus Δ(τ_2, b_2) \boxplus \cdots \boxplus Δ(τ_r, b_r)\,, $$ where $Δ(τ_i,b_i)$'s are Speh representations in the discrete spectrum of $\mathrm{GL}_{a_ib_i}(\mathbb{A})$ with $τ_i$'s being unitary cuspidal representations of $\mathrm{GL}_{a_i}(\mathbb{A})$, and $n = \sum_{i=1}^r a_ib_i$. Endoscopic lifting i…
The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $Γ$ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle fo…
Let $\mathscr{L}(S^1)$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^1$ by its rational points. We give a complete description of the structure of $\mathscr{L}(S^1)$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^1$, which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $\mathscr{L}(S^1)$ is 2. Also we characterize the points on $S^1$ whose Lagrange numbers are less than 2 in terms of Romik's di…
Let $G$ be a split reductive group over a number field $F$. We consider the computation of the inner product of two $K$-spherical pseudo Eisenstein series of $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following a classical approach initiated by Langlands. We show that only the singularities of the intertwining operators due to the poles of the completed Dedekind zeta function $Λ_F$ contribute to the spectrum, while the singularities caused by the zeroes of $Λ_F$ do not contribute to any of the iterated residues which arise as a result of the necessary contour shifts. In the co…
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to th…
We investigate two inequalities of Bugeaud and Laurent, each involving triples of classical exponents of Diophantine approximation associated to $\ux\in\mathbb{R}^n$. We provide a complete description of parameter triples that admit equality for suitable $\ux$, which turns out rather surprising. For $n=2$ our results agree with work of Laurent. Moreover, we establish lower bounds for the Hausdorff and packing dimensions of the involved $\ux$, and in special cases we can show they are sharp. Proofs are based on the variational principle in parametric geometry of numbers, we enclose sketches of …
The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms of the Hausdorff-dimension of suitable subsets of the set of bounded geodesic rays. We establish estimates on the Hausdorff-dimension of these subsets and thereby obtain non-trivial bounds for the dimension functions. Moreover we discuss the property of B being an absolute winning set, therefore satisfying a remarkable rigidity. Finally, we apply the obtaine…
Following the regularization method presented by Zydor, we study in this paper the regularized linear periods of square-integrable automormphic forms on $\mathrm{GL}_{2n}(\mathbb{A}_F)$, where $F$ is a number field and $\mathbb{A}_F$ its ring of adeles. We obtain a formula that expresses the regularized period of a noncuspidal, square-integrable automorphic form in terms of degenerate Whittaker functions in an inductive manner. As a consequence we characterize irreducible automorphic representations in the discrete spectrum of $\mathrm{GL}_{2n}(\mathbb{A})$ that are distinguished by $\mathrm{G…
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are clo…
Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer and results by D. Roy, we show that German's transference inequalities between the two most classical exponents of uniform Diophantine approximation are optimal. Further, we establish that the spectrum of the $n$ uniform exponents of Diophantine approximation in dimension $n$ is a subset of $\mathbb{R}^n$ with non empty interior. Thus, no Jarník-type relation holds between them.
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.
We provide a few natural applications of the analytic newvectors, initiated in \cite{JN} arXiv:1911.01880, to some analytic questions in automorphic forms for $\mathrm{PGL}_n(\mathbb{Z})$ with $n\ge 2$, in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato--Tate measure, and a second moment estimate of the central $L$-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribut…
The aim of the present article is to render the spectral theory of mean values of automorphic $L$-functions -- in a unified fashion. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere. In this new version we added details to all aspects of our argument especially for t…
For automorphic representations in the nontempered cuspidal spectrum of $\mathrm{SO}_5$, we prove the refined Gan-Gross-Prasad conjecture by establishing a precise Bessel period formula, in which the square of the global Bessel period is expressed as an Euler product of regularized local Bessel period integrals of matrix coefficients.
We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.
The field of formal Laurent series is a natural analogue of the real numbers, and mathematicians have been translating well-known results about rational approximations to that setting. In the framework of power series over the rational numbers, we define and study the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to representation by indefinite binary quadratic forms. We compute both spectra explicitly, and show that they coincide and exhibit no gaps, contrary to what happens over the reals.
We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum, depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, uniform lower and upper bounds are established for the Robin-Neumann gaps in terms…
We show that every irreducible representation in the discrete automorphic spectrum of GL(n) admits a non vanishing mixed (Whittaker-symplectic) period integral. The analog local problem is a study of models first considered by Klyachko over a finite field. Locally, we show that for a p-adic field F every irreducible, unitary representation of GL(n,F) has a Klyachko model.
