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This is the Past Exam of Math Tripos which includes Probabilistic Combinatorics, Population Dynamics, Physical Cosmology, Phase Transitions and Collective Phenomena etc. Key important points are: Modular Forms, Standard Fundamental Domain, Algebraically Independent, Fourier Expansions, Standard Analytic, Riemann-Hurwitz, Primitive Dirichlet, Holomorphic Continuation, Euler Product, Holomorphic
Typology: Exams
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Tuesday 5 June 2001 1.30 to 4.
Attempt at most THREE questions. The questions carry equal weight.
1 Describe the standard fundamental domain D for the action of SL 2 (Z) on the upper half plane H and show that it is, indeed, a fundamental domain. Determine all elliptic fixed points of SL 2 (Z) in D and their stabilizers.
2 Show that the ring of modular forms on SL 2 (Z) is generated by the holomorphic Eisenstein series of weights 4 and 6, which are algebraically independent over C.
3 Define the Hecke operators T (n) acting on Sk(SL 2 (Z)) and determine their action on Fourier expansions. If f ∈ Sk(SL 2 (Z)) satisfies f |T (n) = λ(n)f for all n > 1, show that L(f, s) admits an appropriate Euler product.
4 Let f =
n> 1 a(n)q
n (^) and g = ∑ n> 1 b(n)q
n (^) be elements of Sk(SL 2 (Z)). Using the
Rankin-Selberg method, express the function F (s) =
n> 1 a(n)b(n)n
−s (^) in terms of f, g
and the non-holomorphic Eisenstein series
E(z, s) = π−sΓ(s)
(c,d^ c,d) 6 =(0∈Z,0)
ys |cz + d|^2 s^
(Re(s) > 1).
Assuming the standard analytic properties of E(z, s) (holomorphic continuation in s to C − { 0 , 1 }; simple poles at 0, 1; invariance under s ←→ 1 − s), deduce similar properties of F (s). If L(f, s) and L(g, s) admit an Euler product, show that the same holds for F (s).
5 Applying the Riemann-Hurwitz formula to the projection X(N ) = Γ(N )\H∗^ −→ SL 2 (Z)\H∗, determine the genus of X(N ) (N > 1).
6 Let f =
n> 1 a(n)q
n (^) ∈ Sk(Γ 0 (N ), ψ) and let χ be a primitive Dirichlet character
modulo D, where (N, D) = 1. Show that fχ =
n> 1 a(n)χ(n)q
n (^) lies in
Sk(Γ 0 (D^2 N ), χ^2 ψ). Determine fχ
in terms of g = f
, where
(f
a b c d
)(z) = (ad − bc)k/^2 (cz + d)−kf
az + b cz + d
Paper 75
