Lecture Notes on Modular Forms Shiyue Li Mathcamp 2018 Acknowledgment: This is a week-long course on modular forms that I taught in Week 2 of Canada/USA Mathcamp 2018. The notes are improved and completed via conver- sations with Aaron and students in the class. Contents 1 What are Modular Forms? 2 1.1 Why Do We Care? . .2 1.2 Modular Group and Action on the Upper Half Plane . .3 1.3 Exercises for Day 1 . .5 1.4 Eisenstein Series . .6 1.5 The q-expansion of Modular Forms . .7 1.6 Exercises for Day 2 . 10 2 Vector Space Structure on Modular Forms of Weight 2k 11 2.1 Exercises for Day 3 . 14 2.2 Ramanujan’s Cusp Form . 15 2.3 The Dimension Formula . 15 3 Back to the End 17 3.1 The Modular Invariant . 17 3.2 Exercises for Day 4 . 19 3.3 Lattices . 20 3.4 Elliptic Functions and Weierstrass } Function . 20 1 1 What are Modular Forms? 1.1 Why Do We Care? Modular forms were considered central to Wiles’ proof of Fermat’s Last Theorem. The history of modular forms along the line of FLT is roughly as follows: • In 1925, Hecke contibuted hugely to the pillar of the theory of modular forms. • During Hitler and the war, most of the celebrated progress made by German math- ematicians in this field except for Eichler, Maass, Petersson, and Rankin were ig- nored. • In 1956, Taniyama (1956) stated a preliminary (slightly incorrect) version of the Taniyama-Shimura Conjecture: Every rational elliptic curve is modular. • In 1967, Weil rediscovered the Taniyama-Shimula conjecture and showed it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve. • In 1986, Frey called attention to the curve y2 = x(x − an)(x − bn), whose solution (if exists) will suggest that the curve is not modular and is the solution to an + bn = cn, n ≥ 3, disproving Fermat’s Last Theorem. Applications of modular forms to other problems in number theory and arithmetic geometry abound. Just to name a few: • Congruent number problem: This ancient open problem is to determine which integers are the area of a right triangle with rational side lengths. There is a po- tential solution that uses modular forms (of weight 3/2) extensively (the solution is conditional on truth of the Birch and Swinnerton-Dyer conjecture, which is not yet known). • Cryptography and Coding Theory. Points counting on elliptic curves over fi- nite fields is crucial for constructing elliptic curve cryptosystems. Computation of modular forms gives efficient algorithms for point counting. • Generating functions for partitions. The generating functions for partitioning an integer can be related to modular forms. The following box contains mysterious stories that we might or might not unfold in this class, but it will give us a general sense of directions. Big Picture: There is a one to one correspondence between the orbit space of SL2(Z) action on H and the isomorphism classes of elliptic curves. That is, 1:1 ∼ SL2(Z) n H !f elliptic curves over Cg/ = . 2 Modular forms are related to the left hand of the correspondence. So let’s start from here. 1.2 Modular Group and Action on the Upper Half Plane Definition 1.1. Let H be the upper half plane of C; that is, H = fz 2 H : z = x + iy, y > 0g. Definition 1.2 (Modular Group). The full modular group or modular group is the group of all matrices of the form 0 1 a b g = @ A with a, b, c, d 2 Z and det(g) = ad − bc = 1. c d This group is called SL2(Z) or special linear group of 2 × 2 matrices over Z. Example 1.3. The matrices 0 1 0 1 0 −1 1 1 S = @ A , and T = @ A 1 0 0 1 are both elements of SL2(Z). In fact, S, T generate SL2(Z) (Exercise). Remark 1.4. The relationship between Z and R is very similar to the relationship of SL2(Z) and SL2(R): Z and SL2(R) are the discrete subgroup of the larger group. Definition 1.5. For a group G and a set S, a group action A is a map from G × S to S, such that • (e, s) 7! s for all s 2 S and e is the identity element; • (gh, s) 7! (g, (h, s)) for all g, h 2 G and s 2 S. 0 1 a b Definition 1.6. For any g 2 SL2(Z), and any point z 2 H, we can let g = @ A send c d az+b z to cz+d . This defines a group action of SL2(Z) on H (Exercise). Definition 1.7. For each element z in H, the orbit of z under the action of G, is defined to be Gz := fgz : g 2 Gg. 3 Figure 1: Fundamental Domain D of H under action of SL2(Z). Definition 1.8. A fundamental domain D of H under the action of G is defined to be the set z 2 H such that for any x 2 H, there exists some x0 2 D and g 2 G such that gx0 = x. In other words, a fundamental domain D of H under the action of SL2(Z) is a set of orbit representatives under the action of SL2(Z). Modular forms are nothing but some “nice” complex functions H! C such that it satisfies some symmetries under the group action SL2(Z). By “nice”, we mean that the functions do not blow up themselves. Definition 1.9 (Meromorphic Functions). Let D be an open subset of C. A function f : D ! C [ f¥g is meromorphic if it is holomorphic except (possibly) at a discrete set S of points in D, and at each a 2 S there is a positive integer n such that (z − a)n f (z) is holomorphic at a. Example 1.10. Are these functions meromorphic? f (x) = 1 (i) x2+1 ; 3 f (z) = z −2z+10 (ii) z5+3z−1 ; (iii) h(z) = ez. Remark 1.11. Every meromorphic functions can be written as ratio of two holomorphic functions but the denominator cannot be constantly 0. Definition 1.12 (Holomorphic Functions). Let D be an open subset of C. A function f : D ! C is holomorphic if f is complex differentiable at every point z 2 D, i.e. for each z 2 D, the limit f (z + h) − f (z) f 0(z) = lim h!0 h 4 exists, where h may approach 0 along any path. Remark 1.13. The existence of a complex derivative in a neighborhood of z0 is a very strong condition, for it implies that any holomorphic function is actually infinitely dif- ferentiable and equal to its own Taylor series (analytic) in a neighborhood of z0. Definition 1.14 (Weakly Modular Function of Weight k). A weakly modular function of weight N 2 Z is a meromorphic function f on H such that for all g 2 SL2(Z) and all z 2 H, we have az + b f (z) = (cz + d)−n f (g(z)) = (cz + d)−n f . cz + d The constant functions are weakly modular of weight 0. In Exercise, we will show that there are no odd-weighted weakly modular functions. For later convenience, we will use 2k, k ≥ 0 for weight 2k and skip all odd weights. We will later show that there are many weakly modular functions of weight 2k, for k a positive integer. 1.3 Exercises for Day 1 Exercise 1.15. If you have not seen the definition of group before, look up the definition of a group and show that group of 2 × 2 matrices with integer coefficients and with determinant 1 form is indeed a group under matrix multiplication. 0 1 a b Exercise 1.16. For any g = @ A 2 SL2(Z), and any z 2 H, what is the imaginary c d part of gz? Exercise 1.17. Show that the map A : SL2(Z) × H 7! H, defined by az + b (g, z) 7! cz + d 0 1 a b where g = @ A 2 SL2(Z) and z 2 H, is a group action. c d Exercise 1.18. Show that S and T (Example 1.3) are the generators of SL2(Z). Exercise 1.19. Every rational function (quotient of two polynomials) is a meromorphic function on C. Exercise 1.20. Show that there cannot be weakly modular functions of odd weights. Exercise 1.21. Show that the differential form of weight 2k, f (z)(dz)2k, is invariant un- der the action of every element of SL2(Z). 5 1.4 Eisenstein Series Recall that we discussed the group action of SL2(Z) and its fundamental domains in the upper half plane H. Thoughout, we work with a chosen fundamental domain 1 D = fz 2 H : jRe(z)j ≤ , jzj ≥ 1g. 2 It is worth mentioning that subgroups of SL2(Z) fixes certain points in H. (Exercise) • i is fixed by the subgroup of SL2(Z) generated by S. 2pi/3 • r = e fixed by the subgroup of SL2(Z) generated by ST. pi/3 • −r = e fixed by the subgroup of SL2(Z) generated by TS. We recall the following definitions. Definition 1.22 (Modular Function). A modular function of weight 2k is a weakly mod- ular function of weight 2k that is meromorphic at i¥. Definition 1.23 (Modular Form).