Gelbart, Stephen S. Automorphic Forms on Adele Groups. (AM), Volume Series:Annals of Mathematics Studies PRINCETON UNIVERSITY PRESS. Automorphic Representations of Adele Groups. We have defined the space A(G, Γ) of auto- morphic forms with respect to an arithmetic group Γ of G (a reductive. Download Citation on ResearchGate | Automorphic forms on Adele groups / by Stephen S. Gelbart | “Expanded from notes mimeographed at Cornell in May of.
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Pretty much the only way to take an automorphic representation and prove that it has an associated Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and adeld it into pieces and pick out the one you want.
Lion, Vergne, The Weil representation, Maslov index and theta series. A large class of them can be constructed as follows. I haven’t taken a course in representation theory beyond representations of finite groups and would be interested in knowing where I can get the required representation theory background before reading the book. AMVolume 82 Joan S. AMVolume 72 John Milnor.
Or does it just mean that modular forms for adeles and modular forms for the complex numbers are slightly different beasts? Bump or Goldfeld will explain the latter, without necessarily touching on the former. Cycles, Transfers, and Motivic Homology Theories. I have a solid background knowledge at the first-year graduate level I’ve passed qualsbut I don’t know what the prerequisites for Gelbart’s book are. And if you want to get into the whole automorphic representations on adeles groups then some knowledge of algebraic groups and representations of reductive algebraic groups.
To get into the Langlands program there’s the book an introduction to the Langlands program google books you could gellbart at.
In fact, while recently the role of Galois representations has been highlighted Langlands sdele, modularity theoremthis is an entirely separate and higher level issue compared with the basic dictionary between modular forms and automorphic representations. Apurv 2, 16 There is adfle an abundance of advanced books on Galois representations and automorphic forms. In fact, my main familiarity with them is in showing the equivalence of ideal-theoretic and idelic class field theory, which is precisely your “toy version,” so I’m quite curious to eventually understand what you talked about.
The Classical Theory 2. This is a finite-level “shadow” of the rich interaction of Hecke theory and the representation theory of adelic groups, a perspective which allows one to “localize” problems gelbat modular forms in a manner that can be difficult to express in purely classical terms. Progress in Mathematics, 6. The Trace Formula for GL 2 Home Questions Tags Users Unanswered. I’ve browsed through the book.
In analytic number theory, one often sees sums over characters – but characters are automorhpic very relevant in representation theory.
Automorphic Forms on Adele Groups. (AM-83), Volume 83
AMVolume 76 John Milnor. Wait, how does a Hecke character give a modular form? Since you mentioned Galois representations, I can briefly discuss the simplest version of the connection there and point you to Diamond and Shurman’s excellent book which discusses modular forms with an aim towards this perspective.
Dispatched from the UK in 4 business days When will my order arrive? More importantly, I have a basic background in the representation theory of finite groups.
It seems like some kind of group theory. But hopefully what I’ve written has helped you out a bit.
The functional equation can be seen as an adelic version of the Poisson summation formula. It is a pity that things like these don’t occur in complex analysis courses. But also the representation theory of reductive Lie groups both the finite-dimensional case and the infinite-dimensional case, automorphid least the “admissible” case.
How is representation theory used in modular/automorphic forms? – MathOverflow
See Terry Gannon’s book. If you were to pick up Bump’s book Automorphic forms and representations he’ll go over some background. Post as a guest Name. When you return to college in the fall, ask any of the many expert number theorists in the math department there.
This gives an inkling of the connection. Here are some ideas which might bear fruit: One goal is to interpret some recent developments in this area, most significantly the theory of Jacquet-Langlands, working out, whenever possible, explicit consequences and connections with the classical theory. David Corwin 6, 6 66