By M. Ram Murty, V. Kumar Murty
ISBN-10: 8132207696
ISBN-13: 9788132207696
Srinivasa Ramanujan used to be a mathematician exceptional past comparability who encouraged many nice mathematicians. there's vast literature on hand at the paintings of Ramanujan. yet what's lacking within the literature is an research that may position his arithmetic in context and interpret it by way of sleek advancements. The 12 lectures via Hardy, brought in 1936, served this goal on the time they got. This ebook provides Ramanujan’s crucial mathematical contributions and offers an off-the-cuff account of a few of the most important advancements that emanated from his paintings within the twentieth and twenty first centuries. It contends that his paintings nonetheless has an effect on many various fields of mathematical learn. This e-book examines a few of these topics within the panorama of 21st-century arithmetic. those essays, in line with the lectures given via the authors concentrate on a subset of Ramanujan’s major papers and exhibit how those papers formed the process smooth arithmetic.
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Preface. - bankruptcy 1. The Legacy of Srinivasa Ramanujan. - bankruptcy 2. The Ramanujan tau functionality. - bankruptcy three. Ramanujan's conjecture and l-adic representations. - bankruptcy four. The Ramanujan conjecture from GL(2) to GL(n). - bankruptcy five. The circle technique. - bankruptcy 6. Ramanujan and transcendence. - bankruptcy 7.
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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 Mathematical Legacy of Srinivasa Ramanujan by M. Ram Murty, V. Kumar Murty
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