Riemann Surfaces - 24F Riemann Surfaces De ne X 0:= f(x;y ) 2 C 2: x 3 y + y3 + x = 0 g. (a) Prove...

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Transcript of Riemann Surfaces - 24F Riemann Surfaces De ne X 0:= f(x;y ) 2 C 2: x 3 y + y3 + x = 0 g. (a) Prove...

  • Part II

    — Riemann Surfaces

    Year

    2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005

  • 96

    Paper 3, Section II

    23F Riemann Surfaces Let Λ be a lattice in C, and f : C/Λ → C/Λ a holomorphic map of complex tori.

    Show that f lifts to a linear map F : C → C. Give the definition of ℘(z) := ℘Λ(z), the Weierstrass ℘-function for Λ. Show that

    there exist constants g2, g3 such that

    ℘′(z)2 = 4℘(z)3 − g2℘(z)− g3.

    Suppose f ∈ Aut(C/Λ), that is, f : C/Λ → C/Λ is a biholomorphic group homomorphism. Prove that there exists a lift F (z) = ζz of f , where ζ is a root of unity for which there exist m,n ∈ Z such that ζ2 +mζ + n = 0.

    Paper 2, Section II

    23F Riemann Surfaces (a) Prove that z 7→ z4 as a map from the upper half-plane H to C\{0} is a covering

    map which is not regular.

    (b) Determine the set of singular points on the unit circle for

    h(z) =

    ∞∑

    n=0

    (−1)n(2n + 1)zn.

    (c) Suppose f : ∆ \ {0} → ∆ \ {0} is a holomorphic map where ∆ is the unit disk. Prove that f extends to a holomorphic map f̃ : ∆ → ∆. If additionally f is biholomorphic, prove that f̃(0) = 0.

    (d) Suppose that g : C →֒ R is a holomorphic injection with R a compact Riemann surface. Prove that R has genus 0, stating carefully any theorems you use.

    Part II, 2019 List of Questions

    2019

  • 97

    Paper 1, Section II

    24F Riemann Surfaces Define X ′ := {(x, y) ∈ C2 : x3y + y3 + x = 0}. (a) Prove by defining an atlas that X ′ is a Riemann surface.

    (b) Now assume that by adding finitely many points, it is possible to compactify X ′

    to a Riemann surface X so that the coordinate projections extend to holomorphic maps πx and πy from X to C∞. Compute the genus of X.

    (c) Assume that any holomorphic automorphism of X ′ extends to a holomorphic automorphism of X. Prove that the group Aut(X) of holomorphic automorphisms of X contains an element φ of order 7. Prove further that there exists a holomorphic map π : X → C∞ which satisfies π ◦ φ = π.

    Part II, 2019 List of Questions [TURN OVER

    2019

  • 96

    Paper 2, Section II

    23F Riemann Surfaces State the uniformisation theorem. List without proof the Riemann surfaces which

    are uniformised by C∞ and those uniformised by C.

    Let U be a domain in C whose complement consists of more than one point. Deduce that U is uniformised by the open unit disk.

    Let R be a compact Riemann surface of genus g and P1, . . . , Pn be distinct points of R. Show that R \ {P1, . . . , Pn} is uniformised by the open unit disk if and only if 2g − 2 + n > 0, and by C if and only if 2g − 2 + n = 0 or −1.

    Let Λ be a lattice and X = C/Λ a complex torus. Show that an analytic map f : C → X is either surjective or constant.

    Give with proof an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.

    Paper 3, Section II

    23F Riemann Surfaces Define the degree of an analytic map of compact Riemann surfaces, and state the

    Riemann–Hurwitz formula.

    Let Λ be a lattice in C and E = C/Λ the associated complex torus. Show that the map

    ψ : z + Λ 7→ −z + Λ is biholomorphic with four fixed points in E.

    Let S = E/ ∼ be the quotient surface (the topological surface obtained by identi- fying z + Λ and ψ(z + Λ) ), and let p : E → S be the associated projection map. Denote by E′ the complement of the four fixed points of ψ, and let S′ = p(E′). Describe briefly a family of charts making S′ into a Riemann surface, so that p : E′ → S′ is a holomorphic map.

    Now assume that, by adding finitely many points, it is possible to compactify S′ to a Riemann surface S so that p extends to a regular map E → S. Find the genus of S.

    Part II, 2018 List of Questions

    2018

  • 97

    Paper 1, Section II

    24F Riemann Surfaces Given a complete analytic function F on a domain G ⊂ C, define the germ of a

    function element (f,D) of F at z ∈ D. Let G be the set of all germs of function elements in G. Describe without proofs the topology and complex structure on G and the natural covering map π : G → G. Prove that the evaluation map E : G → C defined by

    E([f ]z) = f(z)

    is analytic on each component of G. Suppose f : R → S is an analytic map of compact Riemann surfaces with B ⊂ S

    the set of branch points. Show that f : R \ f−1(B) → S \B is a regular covering map. Given P ∈ S \B, explain how any closed curve in S \B with initial and final points

    P yields a permutation of the set f−1(P ). Show that the group H obtained from all such closed curves is a transitive subgroup of the group of permutations of f−1(P ).

    Find the group H for the analytic map f : C∞ → C∞ where f(z) = z2 + z−2.

    Part II, 2018 List of Questions [TURN OVER

    2018

  • 93

    Paper 2, Section II

    21F Riemann Surfaces Let f be a non-constant elliptic function with respect to a lattice Λ ⊂ C. Let P be

    a fundamental parallelogram whose boundary contains no zeros or poles of f . Show that the number of zeros of f in P is the same as the number of poles of f in P , both counted with multiplicities.

    Suppose additionally that f is even. Show that there exists a rational function Q(z) such that f = Q(℘), where ℘ is the Weierstrass ℘-function.

    Suppose f is a non-constant elliptic function with respect to a lattice Λ ⊂ C, and F is a meromorphic antiderivative of f , so that F ′ = f . Is it necessarily true that F is an elliptic function? Justify your answer.

    [You may use standard properties of the Weierstrass ℘-function throughout.]

    Paper 3, Section II

    21F Riemann Surfaces Let n > 2 be a positive even integer. Consider the subspace R of C2 given by the

    equation w2 = zn − 1, where (z, w) are coordinates in C2, and let π : R → C be the restriction of the projection map to the first factor. Show that R has the structure of a Riemann surface in such a way that π becomes an analytic map. If τ denotes projection onto the second factor, show that τ is also analytic. [You may assume that R is connected.]

    Find the ramification points and the branch points of both π and τ . Compute the ramification indices at the ramification points.

    Assume that, by adding finitely many points, it is possible to compactify R to a Riemann surface R such that π extends to an analytic map π : R→ C∞. Find the genus of R (as a function of n).

    Part II, 2017 List of Questions [TURN OVER

    2017

  • 94

    Paper 1, Section II

    23F Riemann Surfaces By considering the singularity at ∞, show that any injective analytic map f : C → C

    has the form f(z) = az + b for some a ∈ C∗ and b ∈ C. State the Riemann–Hurwitz formula for a non-constant analytic map f : R→ S of

    compact Riemann surfaces R and S, explaining each term that appears.

    Suppose f : C∞ → C∞ is analytic of degree 2. Show that there exist Möbius transformations S and T such that

    SfT : C∞ → C∞

    is the map given by z 7→ z2.

    Part II, 2017 List of Questions

    2017

  • 81

    Paper 3, Section II

    20H Riemann Surfaces Let f be a non-constant elliptic function with respect to a lattice Λ ⊂ C. Let

    P ⊂ C be a fundamental parallelogram and let the degree of f be n. Let a1, . . . , an denote the zeros of f in P , and let b1, . . . , bn denote the poles (both with possible repeats). By considering the integral (if required, also slightly perturbing P )

    1

    2πi

    ∂P z f ′(z) f(z)

    dz,

    show that n∑

    j=1

    aj − n∑

    j=1

    bj ∈ Λ.

    Let ℘(z) denote the Weierstrass ℘-function with respect to Λ. For v,w /∈ Λ with ℘(v) 6= ℘(w) we set

    f(z) = det

     

    1 1 1 ℘(z) ℘(v) ℘(w) ℘′(z) ℘′(v) ℘′(w)

      ,

    an elliptic function with periods Λ. Suppose z 6∈ Λ, z − v 6∈ Λ and z − w 6∈ Λ. Prove that f(z) = 0 if and only if z+ v+w ∈ Λ. [You may use standard properties of the Weierstrass ℘-function provided they are clearly stated.]

    Paper 2, Section II

    21H Riemann Surfaces Suppose that f : C/Λ1 → C/Λ2 is a holomorphic map of complex tori, and let πj

    denote the projection map C → C/Λj for j = 1, 2. Show that there is a holomorphic map F : C → C such that π2F = fπ1.

    Prove that F (z) = λz + µ for some λ, µ ∈ C. Hence deduce that two complex tori C/Λ1 and C/Λ2 are conformally equivalent if and only if the lattices are related by Λ2 = λΛ1 for some λ ∈ C∗.

    Part II, 2016 List of Questions [TURN OVER

    2016

  • 82

    Paper 1, Section II

    22H Riemann Surfaces (a) Let f : R → S be a non-constant holomorphic map between Riemann surfaces.

    Prove that f takes open sets of R to open sets of S.

    (b) Let U be a simply connected domain strictly contained in C. Is there a conformal equivalence between U and C? Justify your answer.

    (c) Let R be a compact Riemann surface and A ⊂ R a discrete subset. Given a non-constant holomorphic function f : R \ A→ C, show that f(R \ A) is dense in C.

    Part II, 2016 List of Questions

    2016

  • 88

    Paper 3, Section II

    19F Riema