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Automorphic Forms and Even Unimodular Lattices - Kneser Neighbors of Niemeier Lattices (Hardcover, 1st ed. 2019)
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Automorphic Forms and Even Unimodular Lattices - Kneser Neighbors of Niemeier Lattices (Hardcover, 1st ed. 2019)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 69
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This book includes a self-contained approach of the general theory
of quadratic forms and integral Euclidean lattices, as well as a
presentation of the theory of automorphic forms and Langlands'
conjectures, ranging from the first definitions to the recent and
deep classification results due to James Arthur. Its connecting
thread is a question about lattices of rank 24: the problem of
p-neighborhoods between Niemeier lattices. This question, whose
expression is quite elementary, is in fact very natural from the
automorphic point of view, and turns out to be surprisingly
intriguing. We explain how the new advances in the Langlands
program mentioned above pave the way for a solution. This study
proves to be very rich, leading us to classical themes such as
theta series, Siegel modular forms, the triality principle,
L-functions and congruences between Galois representations. This
monograph is intended for any mathematician with an interest in
Euclidean lattices, automorphic forms or number theory. A large
part of it is meant to be accessible to non-specialists.
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