By G. Lorentz

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54 4 The Ramanujan Conjecture from GL(2) to GL(n) We would like to determine its behaviour as y tends to infinity. To do this, we can apply Laplace’s saddle point method: if f has two continuous derivatives, with f (0) = f (0) = 0 and f (0) > 0, and f is increasing in [0, A], then A I (x) := e−xf (t) dt ∼ 0 π 2xf (0) as x tends to infinity and provided that I (x0 ) exists for some x0 . A slightly generalized version of this says that if g is continuous on [0, A], then A g(t)e−xf (t) dt ∼ g(0) 0 π .

We must however understand that much of this work seems to be against the background of two world wars and there was no one giving us the “bigger picture”. In his book, 44 4 The Ramanujan Conjecture from GL(2) to GL(n) Lang [106] wrote, “Partly because of Hitler and the war, which almost annihilated the German school of mathematics, and partly because of the great success of certain algebraic methods of Artin, Hasse, and Deuring, modular forms and functions were to a large extent ignored by most mathematicians for about thirty years after the 1930s.

The most spectacular is the 1965 paper of Selberg [180] where he discusses the spectral theory of the Laplace operator and connects it with estimates for τ (n). In the same paper, he formulates the now celebrated Selberg eigenvalue conjecture of which we shall say more later. ” Indeed, if one looks at the Ramanujan conjecture for general Hecke eigenforms, then in 1954, Eichler, Shimura and Igusa solved it for the case k = 2 by noting that if we consider Γ0 (N ) = γ = a c b ∈ SL2 (Z), c ≡ 0 (mod N ) d then Γ0 (N)\h, suitably compactified, has the structure of a Riemann surface and consequently, can be identified as the C-locus of a curve.

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