Modular Forms (MA4H9) Marc Masdeu Copyright c 2014 Marc Masdeu HTTP://WWW.WARWICK.AC.UK/MMASDEU/ Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this file except in compliance with the License. You may ob- tain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. Contents 1 Modular Forms for SL2(Z) .....................................5 1.1 What can modular forms do for us5 1.1.1 Class numbers.................................................5 1.1.2 An example: a modular form of level 11 .............................5 1.2 Ode to the upper half-plane6 1.3 The modular group7 1.4 Fourier expansions of Eisenstein series9 1.5 Valence formula 10 1.6 A product formula for D(z) 12 1.6.1 The Dedekind-eta function...................................... 12 2 Modular Forms on Congruence Groups ...................... 15 2.1 Congruence subgroups 15 2.2 Fourier expansion at infinity 16 2.3 Expansions at cusps 16 3 Complex tori ............................................... 19 3.1 Lattices and tori 19 3.2 Tori and elliptic curves 21 3.2.1 Meromorphic functions on C=L ................................... 21 3.2.2 The Weierstrass Ã-function....................................... 21 Bibliography ............................................... 25 Index ...................................................... 27 What can modular forms do for us Class numbers An example: a modular form of level 11 Ode to the upper half-plane The modular group Fourier expansions of Eisenstein series Valence formula A product formula for D(z) The Dedekind-eta function 1. Modular Forms for SL2(Z) 1.1 What can modular forms do for us In order to motivate the study of modular forms, we give two example results whose proof relies on the existence of these objects. 1.1.1 Class numbers Let D < 0 be the discriminant of an imaginary quadratic field, and let h(D) the class number (the size of the class group). This number is interesting, for a famous problem studied by Gauss is to know when h(D) = 1 (there are only finitely many such), or in general, list all D for which h(D) = h0 for a fixed h0. Here is a simple modular form: ¥ 2 q(z) = ∑ qn = 1 + 2q + 2q4 + 2q9 + ; q = e2piz: n= ¥ ··· − where z H = z C: Á(z) > 0 . Note that q < 1 so the above makes sense. Take its cube: 2 f 2 g j j ¥ H(z) = q(z)3 = 1 + 6q + 12q2 + 24q163 + 168q251 + = ∑ r(n)qn: ··· ··· ··· n=0 Lemma 1.1.1 If D ∼= 3 (mod 8), then 24h(D) = r(D). There are similar results to the above for other D, the point here is that this gives an easy way to calculate h(D)! 1.1.2 An example: a modular form of level 11 Consider another modular form: ¥ ¥ f (z) = q ∏(1 qn)2(1 q11n)2 = q 2q2 q3 + 2q4 + q5 + 2q6 + = ∑ a(n)qn: n=1 − − − − ··· n=1 6 Chapter 1. Modular Forms for SL2(Z) Theorem 1.1.2 1. a(nm) = a(n)a(m) whenever (n;m) = 1. 2. a(p) 2pp for all prime p. j j ≤ Consider the equation: Y 2 +Y = X3 X2 10X 20; − − − and let N(p) be the number of solutions in Fp. An easy heuristic reasoning leads us to think that N(p) p. ' Theorem 1.1.3 — Hasse. p N(p) 2pp. j − j ≤ It turs out that a(p) = p N(p) for all p, so again we can easily calculate (from f ) what is − N(p) for all p. We say in this case that E “is modular”. The book [DS05] (but not this course) explains how to attach an elliptic curve to a modular form (this is called “Eichler–Shimura”). It is much harder to reverse this process, and this is what A.Wiles did in order to prove Fermat’s Last Theorem. 1.2 Ode to the upper half-plane This section introduces the seemingly innoccuous upper half-plane H. Definition 1.2.1 The upper half-plane H is the set of complex numbers with positive imagi- nary part: H = z = x + iy Á(z) > 0 : f j g The upper half-plane appears in the classification of Riemann surfaces: there are only three of them which are simply connected which are the complex plane, the complex sphere, and H. The general linear group GL2(R) consists of all 2 2 invertible matrices with entries in R. It + × constains the subgroup GL2 (R) of matrices with positive determinant. The special linear group + a b + SL2(R) GL (R) consists of those matrices with determinant 1. For g = GL (R) and ⊂ 2 c d 2 2 z H, define gz as: 2 a b az + b gz = z = : (1.1) c d cz + d + Lemma 1.2.1 Let g = a b GL (R). Then: c d 2 2 det(g)Á(t) Á(gt) = : ct + d 2 j j Proof. Exercise (direct computation). + Corollary 1.2.2 GL2 (R) acts on the left on H. Note that the determinant gives a decomposition + GL (R) = SL2(R) R; 2 × l 0 and since the scalar matrices (those of the form 0 l ) act trivially on H, from now on we will 1 0 restrict our attention to SL2(R). In fact, since the scalar matrix −0 1 belongs to SL2(R), the 1.3 The modular group 7 above action on H factors through PSL2(R) = SL2(R)= 1 , which is called the projective special linear group. {± g From this action we can deduce a right action on functions on H, by precomposing: ( f g)(z) = f (gz): j We will need slightly more general actions on functions, but before we introduce a piece of notation that will later prove useful. Definition 1.2.2 The automorphy factor is the function j : SL2(R) H C × ! given by: a b j(g;z) = cz + d; g = : c d The following lemma gives a very interesting property of the automorphy factor, sometimes referred to as the cocycle relation: Lemma 1.2.3 For every g1, g2 in SL2(Z) and for every z H we have: 2 j(g1g2;z) = j(g1;g2z) j(g2;z): Finally, we define an action of SL2(R) on functions f : H C, for each k Z. ! 2 Definition 1.2.3 The weight-k slash operator is defined as k ( f kg)(z) = j(g;z)− f (gz): (1.2) j The cocycle property implies that if f is a function, then: f k(g1g2) = ( f kg1) kg2; g1;g2 SL2(R): j j j 8 2 That is, for each k the weight-k slash operator defines an action of SL2(R) on functions on the upper-half plane. 1.3 The modular group Let G = SL2(Z)SL2(R) be the subgroup of matrices with entries in Z (and determinant 1), which of course still acts on functions as we have seen. Definition 1.3.1 A holomorphic function f : H C is called weakly-modular of weight ! k Z for G if: 2 f (g z) = j(g;z)k f (z); g G: (1.3) · 8 2 We will need an extra analytic property to define modular forms for G. For now, note that: d(g z) · = j(g;z) 2; dz − so we can rewrite the weakly-modular property by asking that the differential f (z)(dz)k=2 is invariant under G. It also shows that if (1.3) holds for g1 and g2, then it also holds for g1g2. 8 Chapter 1. Modular Forms for SL2(Z) Lemma 1.3.1 G is generated by: 1 1 0 1 T = ; S = : 0 1 1− 0 The lemma and the observation before it imply that for f to be weakly-modular it is enough to check (1.3) for T and S: f (z + 1) = f (z); f ( 1=z) = zk f (z): − Example 1.1 — Eisenstein Series. For k 3, define ≥ k Gk(z) = ∑ 0 (mz + n)− : (m;n) Z2 2 Lemma 1.3.2 Gk converges absolutely for all z, and uniformly on compact subsets. Therefore we may rearrange the series, and deduce that Gk is weakly-modular. R If k is odd then Gk(z) is identically zero. For even k, we have: 1 lim Gk(z) = = 2z(k): z i¥ ∑ nk ! k Z 2 The transformation property (rather, the fact that f (z+1) = f (z)) implies that f has a Fourier expansion. Another way to think about it is that there is a holomorphic map: 2piz exp: H 0 < q < 1 ; z q = e : ! f j j g 7! If f is holomorphic and 1-periodic, then we can define g(q) = f (z). That is, we may define: logq g(q) = f ; 2pi where we may choose any branch of the logarithm because of the periodicity of f . The function g is holomorphic on D0, and thus it has a Laurent expansion ¥ g(q) = ∑ a(n)qn: n= ¥ − Therefore f has an expansion ¥ f (z) = ∑ a(n)e2pinz: n= ¥ − Definition 1.3.2 We say that f is meromorphic at ¥ (respectively holomorphic at ¥) if n f (z) = ∑n n0 a(n)q (respectively if in addition n0 = 0). ≥ Note that checking that f is holomorphic at infinity is the same as checking that f (z) is bounded as z approaches i¥.