METAPLECTIC EISENSTEIN SERIES

In this new version the Boltzmann weights are given by number-theoretic quantities in contrast to the standard situation in mathematical physics, the new Boltzmann weights are not field free. Permanent link to this document https: In addition, the principal investigator and his collaborators Chinta and Hoffstein have given new conjectures concerning the residues of these Eisenstein series, conjectures which they have verified for the rational function field and which it should be possible to test for higher degree function fields. He has also co-advised one doctoral students, Ting-Fang Lee, who just defended her dissertation, which included work on multiple Dirichlet series over function fields, and has served as postdoctoral advisor to Lei Zhang, who is now working in the area. Metaplectic Eisenstein series and the Bump-Hoffstein conjecture. In the series under construction, when the primes are put together to make a global object, the different primes interact as one takes their product, rather than combining independently.

Moreover, they have shown that an alternative expession may be given using a new version of the 6-vertex model which appears in statistical mechanics. It is also proposed to develop further connections to the theory of quantum groups. Permanent link to this document https: This proposal seeks to exhibit a new class of global objects, also reflective of a local-to-global principle, but in a new way. More by Toshiaki Suzuki Search this author in: Other groups and their modular and automorphic forms several variables Secondary: For the one-fold cover, they have shown that the Yang-Baxter equation may be used to establish properties of the series, but for higher degree covers methods from number theory and polytope geometry appear to be required.

Moreover this connection may be used to study the L-functions that arise and to suggest conjectures for more general L-functions.

[] Whittaker Coefficients of Metaplectic Eisenstein Series

This proposal focusses on understanding the Dirichlet series that arise when the automorphic representation is one on a metaplectic cover of the adelic points of a reductive group. This program describes a family of Eulerian L-functions, each attached to an automorphic representation on the adelic points of a reductive group G and serues finite dimensional complex analytic representation of the L-group of G. The expressions, remarkably, involve crystal graphs.

Other groups and their modular and automorphic forms several variables Secondary: He has also co-advised one doctoral students, Ting-Fang Lee, who just defended her dissertation, which included work on multiple Dirichlet series over function fields, and has served as postdoctoral advisor to Lei Zhang, who is now working in the area.

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Google Scholar Project Euclid. Representation-theoretic methods; automorphic representations over local and global fields Citation Suzuki, Toshiaki. Further links to lattice models have also been established. The recent prior work of the principal investigator and his collaborators has shown that even in the simplest case — Eisenstein series induced from the Borel — the Whittaker coefficients of metaplectic Eisenstein series have a remarkably rich structure, and are related to the theory of crystal graphs, which also arise in the study of quantum groups.

A Whittaker coefficient is a generalization of a Fourier coefficient. You do not have access to this content. It is also proposed to develop further connections to the theory of quantum groups.

This project has established new links between number theory and mathematical physics, links which also involve representation theory and combinatorics.

Additional work is in progress concerning other classes of Eisenstein series and their coefficients. A fundamental theme of modern number theory has been that the Whittaker coefficients of certain Eisenstein series, attached to the adelic points of metxplectic groups, may be expressed in terms of number theoretic objectsLanglands L-functionsand metaolectic encode important arithmetic information.

These covers arise naturally and arithmetically and so one eisenstfin expect that the Whittaker coefficients of Eisenstein series on such groups will also be of number theoretic significance. Representation-theoretic methods; automorphic representations over local and global fields.

Many problems in modern number theory are of a local-to-global nature: This proposal seeks to exhibit a new class of global objects, also reflective of a local-to-global principle, but in a new way.

The Langlands program is fundamental to modern number theory. More by Toshiaki Suzuki Search this author in: Read more about accessing seeries Buy article.

In the series under construction, when the primes are put together to make a serjes object, the different primes interact as one takes their product, rather than combining independently. These properties may be reinterpreted as a new instance of commuting transfer matrices.

Suzuki : Metaplectic Eisenstein series and the Bump-Hoffstein conjecture

The investigation of metaplectic Whittaker coefficients has the potential to provide a rich new family of objects of number-theoretic interest. Fourier coefficients of automorphic forms 11F For the one-fold cover, they have shown that the Yang-Baxter equation may be used to establish properties of the series, but for higher degree covers methods from number theory and polytope geometry appear to be required. eisenstwin

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An Eisenstein series is a function with certain invariance properties created by an averaging process. For example, going back to Riemann and Dirichlet, one takes information at p and encodes it in a function of a variable s, and then multiplies these functions to get a new function of s whose properties reflect all local properties metapelctic whose behavior in s is related to the problem that one began with often in subtle ways.

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Mathematics > Number Theory

On the mathematical physics and representation theory side, the objects of study are crystal graphs, which seriee related to representations of quantum groups. Zentralblatt MATH identifier Download Email Please enter a valid email address. The formulas for the Whittaker coefficients on covers are also linked, for the one-fold cover, to certain constructions in combinatorics, and it appears likely that this research will lead to some new results in this area.

In this new version the Boltzmann weights are given by number-theoretic quantities in contrast to the standard situation in mathematical physics, the new Boltzmann weights are not field metplectic. Dates First available in Project Euclid: You have partial access to this content. Langlands was led to his conjectures about these L-functions by the study of the constant and Whittaker coefficients of Eisenstein series, as these coefficients can be expressed in terms of such L-functions.

If you have a personal subscription to this journal, then please login. The principal investigator and his collaborators Brubaker and Bump have established this in a number of general metalpectic, showing eisensetin for classes of certain Eisenstein series attached to the Borel subgroup the coefficients may be expressed as infinite sums of Gauss sums and degenerate Gauss sums.