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The authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of \\mathrm{GL}_n over \\mathbb Q of any given infinitesimal character, for essentially all n \\leq 8. For this, they compute the dimensions of spaces of level 1 automorphic forms for certain semisimple \\mathbb Z-forms of the compact groups \\mathrm{SO}_7, \\mathrm{SO}_8, \\mathrm{SO}_9 (and {\\mathrm G}_2) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of \\mathrm{GL}_n with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.

Let

We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Let \\pi be the automorphic representation of \\textrm{GSp}_4(\\mathbb{A}) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and \\tau be an arbitrary cuspidal, automorphic representation of \\textrm{GL}_2(\\mathbb{A}). Using Furusawa's integral representation for \\textrm{GSp}_4\\times\\textrm{GL}_2 combined with a pullback formula involving the unitary group \\textrm{GU}(3,3), the authors prove that the L-functions L(s,\\pi\\times\\tau) are \"nice\". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations \\pi have a functorial lifting to a cuspidal representation of \\textrm{GL}_4(\\mathbb{A}). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of \\pi to a cuspidal representation of \\textrm{GL}_5(\\mathbb{A}). As an application, the authors obtain analytic properties of various L-functions related to full level Siegel cusp forms. They also obtain special value results for \\textrm{GSp}_4\\times\\textrm{GL}_1 and \\textrm{GSp}_4\\times\\textrm{GL}_2.

We introduce a class of multilinear singular integral forms

We consider an

We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least For real weights that are not an integer at least A tool in establishing these results is the relation to cohomology groups with values in modules of “analytic boundary germs”, which are represented by harmonic functions on subsets of the upper half-plane. It turns out that for integral weights at least

We construct explicit isomorphisms between spaces of Maass wave forms and cohomology groups for discrete cofinite groups In the case that We introduce the concepts of For spaces of Maass cusp forms we also describe isomorphisms to parabolic cohomology groups with smooth coefficients and standard cohomology groups with distribution coefficients. We use the latter correspondence to relate the Petersson scalar product to the cup product in cohomology.

Let F be a non-Archimedean local field. Let \\mathcal{W}_{F} be the Weil group of F and \\mathcal{P}_{F} the wild inertia subgroup of \\mathcal{W}_{F}. Let \\widehat {\\mathcal{W}}_{F} be the set of equivalence classes of irreducible smooth representations of \\mathcal{W}_{F}. Let \\mathcal{A}^{0}_{n}(F) denote the set of equivalence classes of irreducible cuspidal representations of \\mathrm{GL}_{n}(F) and set \\widehat {\\mathrm{GL}}_{F} = \\bigcup _{n\\ge 1} \\mathcal{A}^{0}_{n}(F). If \\sigma \\in \\widehat {\\mathcal{W}}_{F}, let ^{L}{\\sigma }\\in \\widehat {\\mathrm{GL}}_{F} be the cuspidal representation matched with \\sigma by the Langlands Correspondence. If \\sigma is totally wildly ramified, in that its restriction to \\mathcal{P}_{F} is irreducible, the authors treat ^{L}{\\sigma} as known. From that starting point, the authors construct an explicit bijection \\mathbb{N}:\\widehat {\\mathcal{W}}_{F} \\to \\widehat {\\mathrm{GL}}_{F}, sending \\sigma to ^{N}{\\sigma}. The authors compare this \"naïve correspondence\" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of \"internal twisting\" of a suitable representation \\pi (of \\mathcal{W}_{F} or \\mathrm{GL}_{n}(F)) by tame characters of a tamely ramified field extension of F, canonically associated to \\pi . The authors show this operation is preserved by the Langlands correspondence.