Modular Forms Andrew Kobin Fall 2019 Contents Contents Contents 0 The Web of Modularity 1 0.1 The Partition Function . .2 0.2 Quadratic Forms . .3 0.3 Elliptic Curves . .4 0.4 Galois Theory and Quantum Gravity . .4 0.5 Some Perspectives on Modular Forms . .5 1 Theta Functions 6 1.1 Poisson Summation . .6 1.2 The Jacobi Theta Function . .8 2 Modular Groups 12 2.1 The Modular Group SL2(Z)........................... 12 2.2 Congruence Subgroups . 12 2.3 Modular Curves . 15 3 Modular Forms 17 3.1 Definitions and Examples . 17 3.2 Modular Forms for SL2(Z)............................ 32 3.3 Petersson Inner Product . 35 4 Hecke Operators 37 4.1 Hecke Operators on SL2(Z)........................... 37 4.2 The Trace Form . 41 4.3 Atkin{Lehner Theory . 43 4.4 Oldforms and Newforms . 48 5 Connections to L-Functions 51 5.1 L-Functions for Hecke Eigenforms . 51 5.2 Modular Forms with Complex Multiplication . 53 5.3 Connection to Elliptic Curves with CM . 55 5.4 Periods and Critical L-Values . 57 5.5 Shimura Correspondence . 60 i 0 The Web of Modularity 0 The Web of Modularity These notes were taken during a course / seminar on modular forms taught by Dr. Ken Ono at the University of Virginia in fall 2019. The reading list includes: Koblitz's Introduction to Elliptic Curves and Modular Forms, Cohen-Stromberg's Modular Forms, Chapter VII in Serre's A Course in Arithmetic, Diamond-Shurman's A First Course in Modular Forms and Stein's Modular Forms: A Computational Approach among many others. The course centered around a list of special topics directly tied to the theory of modular forms. Some of these topics include: (1) The theory of elliptic functions and harmonic Maas forms (2) Classical theory of complex multiplication and its connections to abelian class field theory (3) Exact combinatorial formulas, e.g. the partition function p(n), values of the Riemann zeta function like ζ(2n), L-functions of an elliptic curve. (4) Quadratic forms { here's an interesting question that turns out to have a natural modular interpretation: which integers can be represented by an arbitrary quadratic form? For example, if the quadratic form is q = x2, the answer is \all perfect squares" of course. For q = x2 + y2, Legendre and Fermat showed that this comes down to whether the integer is divisible by primes congruent to 3 (mod 4), and this can alternatively be phrased in terms of class field theory. For q = x2 + y2 + 10z2, the question is wide open. None of the odd integers n = 3; 7 and 2719 are representable by q, and in fact 2719 is the largest known non-example. Further: Theorem 0.0.1 (Soundararajan). Assuming the Generalized Riemann Hypothesis, 2719 is the last odd integer not of the form x2 + y2 + 10z2. (5) Modularity of elliptic curves (6) Modular forms mod p (7) Brauer's Problem 19: given a finite simple group G and a fixed prime p, is there a complex irreducible representation ρ : G ! GL(V ) which remains irreducible mod p? (8) Thompson's Conjecture: let G be a finite group and K(G) the number field obtained by adjoining the entries of the character table for G to Q. Thompson proved that if n > 24 and m∗ := (−1)(m−1)=2m for m odd, then p ∗ K(An) = Q(f p j odd primes p < n; p 6= n − 2g): He then conjectured that if Π is a finite set of odd primes, then for n sufficiently large (depending on Π), KΠ(An) = Q(fχ j χ in the char. table for An restricted to g 2 An with order p; p 2 Πg) p is equal to Q(f p∗ j p 2 Π; p 6= n − 2g). 1 0.1 The Partition Function 0 The Web of Modularity Loosely speaking, a modular form is a function f : h ! C satisfying az + b a b An invariance property: f = (cz + d)kf(z) for all lying in a fixed cz + d c d subgroup Γ ≤ SL2(Z) and z 2 h. The number k is called the weight of f, while (cz + d)k is called the factor of automorphy. Analytic, meromorphic or real analytic conditions. In the next few sections, we introduce some classical number theory problems that can be elucidated with the theory of modular forms. In some sense these are the foundations of the theory, as many modern techniques were developed with one of these applications in mind. 0.1 The Partition Function Recall that a partition of an integer n is any nonincreasing sequence of positive integers which sum to n. The partition function p : N ! N is defined by p(n) = the number of partitions of n. It turns out that all of the irreducible representations of every finite group can be explicitly constructed from the combinatorial properties of p(n). For example, if n = 4 there are are 5 partitions: (4); (3; 1); (2; 2); (2; 1; 1); (1; 1; 1; 1). Modular forms enter when one tries to compute the generating function for p(n). Lemma 0.1.1 (Euler). The generating function for p(n) is 1 1 X Y 1 p(n)qn = : 1 − qn n=0 n=1 The right side of this equation is what is known as an `Euler product'. This simple, elegant formula has been a testing ground for the theory of modular forms. For example: The first Hecke operators were defined for this function. The circle method was originally developed with this function in mind. Early examples of Galois representations were computed for this example. Many of the earliest theorems about p(n) are due to Ramanujan, including: Theorem 0.1.2 (Ramanujan). For all n ≥ 0, p(5n + 4) ≡ 0 (mod 5) p(7n + 5) ≡ 0 (mod 7) p(11n + 6) ≡ 0 (mod 11): These were later proven to be the only examples of such congruences. Theorem 0.1.3 (Ahlgren-Boylan). If ` is prime and p(`n + b) ≡ 0 (mod `) for some b 2 Z and all n ≥ 0, then (b; `) = (4; 5); (5; 7) or (6; 11). 2 0.2 Quadratic Forms 0 The Web of Modularity On the other hand: Theorem 0.1.4 (Ono). For any (Q; 6) = 1, there are infinitely many non-nested arithmetic progressions An + B such that p(An + B) ≡ 0 (mod Q) for all n ≥ 0: For example, p(4063467631n + 30064597) ≡ 0 (mod 31) for all n ≥ 0. The asymptotic size of p(n) has also been known for decades: 1 p Theorem 0.1.5 (Hardy-Ramanujan). p(n) ∼ p eπ 2n=3. 4n 3 1 All of these results can be accessed by studying the meromorphic, weight − 2 modular 1 Y 1 function P (z) = q−1=24 , where q = q(z) = e2πiz. Unfortunately, this function has 1 − qn n=1 a a pole at \i1" and at every rational number b 2 Q ⊆ C. 0.2 Quadratic Forms Let v 2 N and define v 2 2 rv(n) = #f(x1; : : : ; xv) 2 Z j x1 + ::: + xv = ng: This is a classical function with many old results known, including: Theorem 0.2.1 (Gauss). Every nonnegative integer is the sum of 3 triangular numbers, i.e. m(m+1) those of the form 2 for some m. Thisp statement reduces to studying r3(8n + 3), namely Gauss showed that r3(8n + 3) = αh(Q( −8n − 3)) for some α ≤ 1, where h is the class number. Theorem 0.2.2 (Lagrange). Every nonnegative integer is the sum of four squares. Theorem 0.2.3 (Landau). For all n ≥ 1, X r4(n) = 8 d: 1≤djn 4-d These results about quadratic forms can be interpreted as statements about the holo- morphic weight v=2 modular form 1 X jjx¯jj2 X n θv(z) = q = rv(n)q : x¯2Zv n=0 Unlike P (z) in the last section, this function has no poles at i1 or any rational number, a and in fact the values of θv(z) as z ! b 2 Q encode important data about values of ζ- and L-functions. 3 0.3 Elliptic Curves 0 The Web of Modularity 0.3 Elliptic Curves An elliptic curve is an algebraic curve E given by an affine model (or Weierstrass equation) y2 = x3 + Ax + B where x3 + Ax + B has no repeated roots. A standard problem in number theory involves counting the rational (or integral) points of E. Theorem 0.3.1 (Mordell-Weil). If K is a number field and E is an elliptic curve, then E(K) is a finitely generated abelian group. ∼ r(E;K) In particular, E(K) = E(K)tors × Z where E(K)tors is a finite abelian group and r(E; K) ≥ 0 is the rank of E(K). The fundamental question in the theory of elliptic curves is: what values of r(E; K) are possible? Example 0.3.2. Consider the elliptic curve E : y2 = x3 − x over Q. Then (0; 0); (1; 0) and (−1; 0) are all order 2 points of E, so E(Q) contains a subgroup isomorphic to Z=2Z×Z=2Z. Define a sequence a(n) = aE(n) by 1 1 X Y a(n)qn = q (1 − q4n)2(1 − q8n)2: n=1 n=1 If p is an odd prime, then a(p) = p − #E(Fp) and in fact, a general theorem of Coates-Wiles implies that r(E; Q) = 0 in this case since the first moment of the generating function above is nonzero.