UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Thursday, 5 June, 2014 1:30 pm to 4:30 pm PAPER 23 MODULAR FORMS Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
1 UNIVERSITY OF CAMBRIDGE 2 a) Briefly explain the construction of the Riemann surface X(T(1)) as a quotient of the extended upper half-plane H [[P1(Q). Describe the points where the map H > X(I(1) is ramified and the ramification index at these points. Prove that X(T(1)) is compact. Let A be the unique normalised cusp form in S12(T(1)) and let E4 be the normalised weight 4 Eisenstein series n>1 E4=1+ 240 > 03(n)q". b) Show that the function j(T) = = 4)-induces a biholomorphism from X(T(1)) to p1 c, and that j(T) has a zero at w = = 1+231 -. [You may assume that A(T) is non- V vanishing on H and has q-expansion q + . . . . ] c) Show that the function j2(T) = 1(21) A(t) induces a biholomorphism from X (To(2)) to Pt. d) Show that (j2 (+) - j2 (w)3 j(T)= . j2 (T)2 Part III, Paper 23
UNIVERSITY OF CAMBRIDGE 3 2 For T E H and q = e2nit the Jacobi 0-function is defined by nIZ 0(T) = > an2. Define another function ø on H by nIZ a) Show that, for T E H, (+) = 0 (1-5) = >(-1)man2. 4? · (IT) = V=0(+). Hence (or otherwise) show that, for positive integers k, ø8k is an element of M4k (To(2)). [You may quote the Poisson summation formula without proof, and assume that To(2) is generated by the matrices 1 0 +(0 1)+(1 0) J - b) Show that S4(To(2)) = {0} and dim M4(To(2)) = 2. [Hint: for f in S4(To(2)) consider the product II allo(2)\SL2(Z) flax, 4.] c) Show that 08 = >(-1)"r8(n)q" n>0 where the integers r8(n) satisfy T8(n) = (-1)"16 ~ (-1)dd3. 0<d?n [It may be helpful to recall that the normalised weight 4 Eisenstein series has q- expansion E4=1+ 240 > 03(n)q".] n>1 Part III, Paper 23 [TURN OVER