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Let E be a cubic étale extension of the rational numbers which is totally real, i.e., E⊗R≃R×R×R. There is an algebraic Q-group SE=Spin8,Ec defined in terms of E, which is semisimple simply-connected of type D4 and for which SE(R) is compact. We let GE denote a certain semisimple simply-connected algebraic Q-group of type D4, defined in terms of E, which is split over R. Then GE×SE maps to quaternionic E8. This latter group has an automorphic minimal representation, which can be used to lift automorhpic forms on SE to automorphic forms on GE. We prove a Siegel–Weil theorem for this dual pair: I.e., we compute the lift of the trivial representation of SE to GE, identifying the automorphic form on GE with a certain degenerate Eisenstein series. Along the way, we prove a few more “smaller\" Siegel–Weil theorems, for dual pairs M×SE with M⊆GE. The main result of this paper is used in the companion paper “Exceptional theta functions and arithmeticity of modular forms on G2\" to prove that the cuspidal quaternionic modular forms on G2 have an algebraic structure, defined in terms of Fourier coefficients.

Let E be a cubic étale extension of the rational numbers which is totally real, i.e., E ⊗ R ≃ R × R × R . There is an algebraic Q -group S E = Spin 8 , E c defined in terms of E , which is semisimple simply-connected of type D 4 and for which S E ( R ) is compact. We let G E denote a certain semisimple simply-connected algebraic Q -group of type D 4 , defined in terms of E , which is split over R . Then G E × S E maps to quaternionic E 8 . This latter group has an automorphic minimal representation, which can be used to lift automorhpic forms on S E to automorphic forms on G E . We prove a Siegel–Weil theorem for this dual pair: I.e., we compute the lift of the trivial representation of S E to G E , identifying the automorphic form on G E with a certain degenerate Eisenstein series. Along the way, we prove a few more “smaller\" Siegel–Weil theorems, for dual pairs M × S E with M ⊆ G E . The main result of this paper is used in the companion paper “Exceptional theta functions and arithmeticity of modular forms on G 2 \" to prove that the cuspidal quaternionic modular forms on G 2 have an algebraic structure, defined in terms of Fourier coefficients.

In a recent work, we found formulas for the Fourier coefficients of automorphic forms of type G 2 : holomorphic Siegel modular forms on Sp 6 that are theta lifts from G 2 c , and cuspidal quaternionic modular forms on split G 2 . We have implemented these formulas in the mathematical software SAGE. In this paper, we explain the formulas of our recent paper and the SAGE implementation. We also deduce some theoretical consequences of our SAGE computations.

Suppose that$G$is a simple reductive group over$\\mathbf{Q}$, with an exceptional Dynkin type and with$G(\\mathbf{R})$quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on$G$along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form$\\unicode[STIX]{x1D703}_{Gan}$on quaternionic$E_{8}$and some applications. The$Sym^{8}(V_{2})$-valued automorphic function$\\unicode[STIX]{x1D703}_{Gan}$is a weight 4, level one modular form on$E_{8}$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic$E_{7},E_{6}$and$G_{2}$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups$G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups$\\operatorname{GSp}_{2n}$.

Suppose that $G$ is a simple reductive group over $\\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\\operatorname{GSp}_{2n}$ .

Se analizan algunos aspectos del tributo de indios y castas en Hispanoamérica, principalmente con base en las fuentes secundarias, con énfasis particular en cómo los estatus fiscales se vinculan con las categorías socioétnicas. A través de una discusión que aborda el contexto en el que nació el tributo, sus símiles en Castilla, su importancia en la creación o fortalecimiento de las distinciones sociales en América, la exención del mismo en casos de participación militar, la importancia del pensamiento económico del siglo XVIII en los argumentos para eliminarlo, las reformas dieciochescas y los procesos que llevaron a su abolición, en estas líneas se (re) plantean algunos acercamientos al tema que ayudan a apreciar la importancia que ha tenido el tributo en el desarrollo de las sociedades hispanoamericanas. This article analyzes some aspects of the tributes made by Indians and castas in Hispanic America and is primarily based on secondary sources, with a particular emphasis on how one's fiscal status was linked to one's socio-ethnic category. Through a discussion of the context of the tribute system's origins, its parallels in Castile, its importance in the creation or strengthening of social distinctions in the Americas, the exemption from tribute due to military service, the importance of 18th Century economic thought in its elimination, the Bourbon reforms and the process that led to their abolition, this article (re) considers the approaches to this subject, highlighting the role of tribute in the development of Hispanic American societies.

A través de una revisión bibliográfica extensiva y la consultación de fuentes primarias complementarias, este artículo analiza las variaciones en los intentos que realizaron los primeros Estados postcoloniales en Hispanoamérica para establecer contribuciones directas proporcionales y cómo las diferentes constituciones fiscales tendieron a debilitar o desmantelarlas. En las regiones con una presencia indígena demográficamente predominante, los diferentes grupos sociales, influidos por sus diversas experiencias como miembros de los estamentos coloniales, rechazaron los impuestos proporcionales, que en poco tiempo se volvieron capitaciones aplicadas principalmente a las poblaciones indígenas, o indígenas y campesinas. A la vez que reproducían, en grande medida, las dinámicas asociadas con el tributo de indios que la monarquía española había cobrado, las nuevas capitaciones influyeron en modificar las categorizaciones sociales (étnicas) del siglo XIX.

We give a Rankin–Selberg integral representation for the Spin (degree eight)$L$-function on$\\operatorname{PGSp}_{6}$that applies to the cuspidal automorphic representations associated to Siegel modular forms. If$\\unicode[STIX]{x1D70B}$corresponds to a level-one Siegel modular form$f$of even weight, and if$f$has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin$L$-function$\\unicode[STIX]{x1D6EC}(\\unicode[STIX]{x1D70B},\\text{Spin},s)$of $\\unicode[STIX]{x1D70B}$.