Even S.'s Algorithmic combinatorics PDF
By Even S.
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Let f be a holomorphic modular form of weight 3 for Γ and consider the 2-form f (z) dz ∧ dw on H × C. It is invariant under the action of Γ Z2 . Indeed, an element a c b , (m, n) d 36 3 Ramanujan’s Conjecture and -Adic Representations acts on f by f az + b az + b d cz + d cz + d ∧d w + mz + n cz + d which is (cz + d)3 f (z)(cz + d)−2 dz ∧ (cz + d)−1 dw = f (z) dz ∧ dw. Because of this invariance, f (z) dz ∧ dw gives rise to a holomorphic 2-form on the quotient and on the surface BΓ . In fact, it is true that all holomorphic 2-forms on BΓ can be obtained in this way in the sense that there is an isomorphism of the space of cusp forms on Γ of weight 3 and holomorphic 2-forms on BΓ .
Though this was first proved by Hardy, his proof was more elaborate. We need only make two observations. The first is that a fundamental domain D for the action of SL2 (Z) on the upper half plane can be taken to be the 1 The Ramanujan Conjectures 43 standard one, namely, |z| ≥ 1, | (z)| ≤ 1/2. In other words, every element in the upper half-plane is SL2 (Z)-equivalent to some element of D, and no two interior elements of D are SL2 (Z)-equivalent. The second observation is that 2 y 12 z = x + iy (z) , is invariant under the action of SL2 (Z) as is easily checked using the modular transformation for .
Using the fact that each of the L-functions L(s, π, rm ) satisfies a functional equation, one can improve the estimate using a classical result of Chandrasekharan and s Narasimhan . This result says that if an ≥ 0 and f (s) = ∞ n=1 an /n is convergent in some half-plane, has analytic continuation for all s except for a pole at s = 1 of order k and satisfies a functional equation of the form Qs (s)f (s) = wQ1−s (1 − s)f (1 − s) where Q > 0 and Γ (αi s + βi ) (s) = i then 2A−1 an = xPk−1 (log x) + O x 2A+1 logk−1 x n≤x where A = i αi .
Algorithmic combinatorics by Even S.