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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Zeta Functions Of Reductive Groups And Their Zeros (Hardcover)
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Zeta Functions Of Reductive Groups And Their Zeros (Hardcover)
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This book provides a systematic account of several breakthroughs in
the modern theory of zeta functions. It contains two different
approaches to introduce and study genuine zeta functions for
reductive groups (and their maximal parabolic subgroups) defined
over number fields. Namely, the geometric one, built up from
stability of principal lattices and an arithmetic cohomology
theory, and the analytic one, from Langlands' theory of Eisenstein
systems and some techniques used in trace formula, respectively.
Apparently different, they are unified via a Lafforgue type
relation between Arthur's analytic truncations and parabolic
reductions of Harder-Narasimhan and Atiyah-Bott. Dominated by the
stability condition and/or the Lie structures embedded in, these
zeta functions have a standard form of the functional equation,
admit much more refined symmetric structures, and most
surprisingly, satisfy a weak Riemann hypothesis. In addition, two
levels of the distributions for their zeros are exposed, i.e. a
classical one giving the Dirac symbol, and a secondary one
conjecturally related to GUE.This book is written not only for
experts, but for graduate students as well. For example, it offers
a summary of basic theories on Eisenstein series and stability of
lattices and arithmetic principal torsors. The second part on rank
two zeta functions can be used as an introduction course,
containing a Siegel type treatment of cusps and fundamental
domains, and an elementary approach to the trace formula involved.
Being in the junctions of several branches and advanced topics of
mathematics, these works are very complicated, the results are
fundamental, and the theory exposes a fertile area for further
research.
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