By Howard Becker
ISBN-10: 0521576059
ISBN-13: 9780521576055
A Polish house (group) is a separable, thoroughly metrizable topological house (group). This ebook is ready activities of Polish teams, in connection with--or from the perspective of--the topic of descriptive set idea. Descriptive set thought is the examine of definable units and capabilities in Polish areas. the fundamental periods of definable units are the periods of Borel, analytic and coanalytic units, and those represent the most subject of the publication, however the authors additionally give some thought to different periods of definable units. it will be a worthwhile publication for all researchers in set thought and comparable components.
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Extra resources for The Descriptive Set Theory of Polish Group Actions
Example text
An e−n/x = O x 1+ . n=1 Thus, for any individual term in the sum, we have an e−n/x = O x 1+ . Choosing x = n, we deduce that an = O(n1+ ). If we let π be an automorphic representation on GL(2) with local parameters, αp , βp , then one can give an intuitive description of the symmetric power Lfunctions attached to π in the following way. Consider, with rm denoting the m-th symmetric power of π , m m−j 1− Lm (s) := L(s, π, rm ) = p j =1 αp j βp −1 ps where we are ignoring the finitely many Euler factors that need to be modified corresponding to the ramified factors.
The subspace S is invariant under both of these actions. Diagonalizing the Hecke action, we get a decomposition S = ⊕f S (f ) into submodules, where each S (f ) is a rank two module over Kf [Gal(Q/Q)], and Kf is a certain number field (the field generated by the Fourier coefficients of f ). 5 Geometric Realization of Modular Forms of Higher Weight 37 These are the Galois representations attached to modular forms of higher weight whose existence had been conjectured by Serre and constructed by Deligne [38].
1007/978-81-322-0770-2_4, © Springer India 2013 39 40 4 The Ramanujan Conjecture from GL(2) to GL(n) wait until 1974 when Deligne proved it as a consequence of his proof of the Weil conjectures. We outline the proofs of these conjectures. As mentioned in the introduction to Chap. 3, the essential property of (z) is that it is a modular form of weight 12 for the full modular group SL2 (Z), which is the group of 2 × 2 matrices with integer entries and determinant 1. This means that for z in the upper half-plane h, we have az + b cz + d = (cz + d)12 (z) a c ∀ b ∈ SL2 (Z).
The Descriptive Set Theory of Polish Group Actions by Howard Becker
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