Show that G2(−1/τ) =τ2G2(τ) +iτ /(4π)

Universit´e Paris Diderot Morningside Center for Mathematics

Exercises (for beginners) on modular forms

1. LetG2(τ) =−1/24 +P∞

n=1σ1(n)qⁿ, whereq=e^2iπτ. Show that G2(−1/τ) =τ²G2(τ) +iτ /(4π).

It is not a straightforward calculation. Letδ(τ) =qQ∞

n=1(1−qⁿ)²⁴, whereq=e^2iπτ. Show that δ⁰/δ=−48iπG2(τ).

Deduce from this equality thatδ(−1/τ) =τ¹²δ(τ) and thatδ∈M₁₂⁰. Finally show thatδ= (E₄³−E₆²)/1728 (= ∆).

2. Letkbe an even integer>2. Denote byM_k (resp. M⁰_k) the space of modular forms (resp. cusp forms) of weightk. Give a formula for the dimensions ofM_k andM⁰_k, according to the class ofkmodulo 12.

3. Show that thatMk is generated by the Eisenstein seriesE4 andE6 as an algebra overC.

4. Denote by H the upper half-plane. Consider the modular function j given by the formula j =E₄³/∆.

Show that j is meromorphic on H ∪P¹(Q) and invariant under SL2(Z). Where are the poles ofj ? Show thatj generates overCthe field of meromorphic functions onH ∪P¹(Q) and invariant under SL2(Z).

5. Letnbe an integer≥1. Show thatT_n(G_k) =σ_k−1(n)G_k. 6. Let S=

0 −1

1 0

and U =

0 −1 1 −1

. LetV_k be the space of complex polynomials of degree≤k−2.

The group GL2(Q) acts onVk by the rule (P, M = a b

c d

)7→P_|M(X) = (ad−bc)^k(cX+d)^−kP((aX+ b)/(cX+d)). LetW_k={P ∈V_k/P +P_|S =P+P_|U +P_|U2 = 0}. LetW_k⁺ (resp. W_k⁻) be the subspace of Vk formed by even (resp. odd) polynomials.

Show thatW₁₂⁻ is one dimensional as a complex vector space and generated by 4X⁹−25X⁷+ 42X⁵− 25X³+ 4X and that W₁₂⁺ is two-dimensional and generated byX¹⁰−1 andX⁸−3X⁶+ 3X⁴−X². 7. Letf ∈ M⁰_k. Denote byr(f) =R∞

0 f(iy)(X−iy)^wi dythe period polynomial off. Show that r(T₂f) =r(f)

|

2 0 0 1

+r(f)

|

2 0 1 1

+r(f)

|

1 1 0 2

+r(f)

|

1 0 0 2

.

Compute the eigenvalues ofT2onW₁₂⁺ andW₁₂⁻. Admit that r(Tnf) = X

a,b,c,d

r(f)

|

a b c d

where (a, b, c, d) runs through the quadruples of integer such thatad−bc=n, anda > c≥0, andd > b≥0.

Devise an algorithm to determine theq-expansion of the basis of newforms ofM⁰_k. 8. Show that iff ∈M_k⁰ admits theq-expansionP∞

n=1a_nqⁿ, one has L(f, s) =Y

p

1

1−app^−s+p^11−2s, wherepruns over prime numbers, if and only iff is a newform.

9. Letf,g∈M_k⁰. Recall that the Petersson scalar product off andgis given by the following formula :

< f, g >=− 1 (2i)^k−1

Z

D

f(z)g(z)dz dz,

whereD is a fundamental domain for the group SL₂(Z). Define a pairing [., .] onV_k by the formula

[X

n

anXⁿ,X

n

bnXⁿ] =X

n

anb_w−n(−1)ⁿ w

n −1

,

so that

[r(f), r(g)] = Z ∞

0

Z ∞

0

f(u)g(v)(v−u)^wdu dv.

Show that the pairing [., .] is symmetric, definite, SL₂(Z)-equivariant, and

−1 0

0 1

antiequivariant. Show then that

< f, g >=− 1 6(2i)^k−1

Z

D₂

f(z)g(z)dz dz,

where D2 is the hyperbolic polygon in the upper half-plane whose summits are −1, 0, 1 and ∞. Set F(z) =Rz

∞f(t)(t−z)¯ ^wdt. Using Stokes’ formula, show that

< f, g >=− 1 6(2i)^k−1

Z

∂D2

F(z)g(z)dz.

Deduce Haberland’s formula

< f, g >=− 1

6(2i)^k−1([r(f)_|T, r(g)]−[r(f), r(g)_|T]), and its refinement

< f, g >=− 1

3(2i)^k−1([r⁺(f)_|T, r⁻(g)]−[r⁺(f), r⁻(g)_|T]).