The Ramanujan Conjecture from GL(2) to GL(n)
Ramanujan’s conjectures regarding the τ-function represent part of a larger spectrum of conjectures, first in the setting of automorphic forms attached to GL(2) and generally to GL(n). In this chapter, we trace this development highlighting what is known
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The Ramanujan Conjecture from GL(2) to GL(n)
1 The Ramanujan Conjectures In retrospect, one sees a progression of ideas from Ramanujan’s work on the τ function to Hecke’s theory of modular forms, and then moving to the representation theoretic perspective, we see this progression linking itself to theories of HarishChandra and Langlands. Ramanujan’s conjecture now sits as a special case of a more comprehensive conjecture in the Langlands program. The survey article [141] provides an excellent introduction to this chain of ideas, and we recommend this to the reader. But the origins of the chain go back to the 1916 paper of Ramanujan which acted as a catalyst for this development. In his epic paper of 1916, Ramanujan [162] considered the function ∞
(z) = q
1 − qn
24
,
q = e2πiz .
n=1
As we have seen, expanding the right-hand side as a power series in q defines the celebrated Ramanujan τ -function: q
∞ n=1
1 − qn
24
=
∞
τ (n)q n .
n=1
In his paper, Ramanujan made three conjectures concerning τ (n): (1) τ (mn) = τ (m)τ (n), for (m, n) = 1, (2) for p prime and a ≥ 1, τ (p a+1 ) = τ (p)τ (p a ) − p 11 τ (p a−1 ), (3) |τ (p)| ≤ 2p 11/2 . The first two conjectures were proved a year later by Mordell [126]. His proof was reproduced by Hardy in his 1936 lectures on Ramanujan’s work delivered at Harvard University. Almost twenty years after Ramanujan’s paper, Hecke [77] published the conceptual framework from which (1) and (2) emerge as special cases of a larger theory, now called Hecke theory. Despite heroic efforts to settle conjecture (3) by such luminaries as Hardy, Kloosterman, Rankin and Selberg, we had to M.R. Murty, V.K. Murty, The Mathematical Legacy of Srinivasa Ramanujan, DOI 10.1007/978-81-322-0770-2_4, © Springer India 2013
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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 a b 12 ∈ SL2 (Z). (21) = (cz + d) (z) ∀ c d cz + d The set of holomorphic functions f : h → C vanishing at infinity and satisfying the modular relation az + b a b k f ∈ SL2 (Z) = (cz + d) f (z) ∀ c d cz + d forms a finite-dimensional vector space Sk over C. Since −I ∈ SL2 (Z), the modular relation shows that Sk = 0 for k odd. For k even, one can show without too much difficulty [183] that dim S2 = 0 and dim Sk = [k/12] if k ≡ 2 (mod 12). Thus, S12 is one-dimensional and spanned by . In fact, 12 is the first value of k for which Sk = 0. Hecke discovered a family of linear transformations Tn of the finite-dimensional vector space Sk . These are now called Hecke operators, though they are nascent in Mordell’s work. To describe these, it is convenient to introduce
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