CHAPTER 1 Basic facts Modular forms and functions play many roles in number theory. Before pro- ceeding to the main topics of this monograph, we begin by recalling (without proof) the fundamental objects and the most basic facts about modular forms. More com- plete details of this introductory material may be found in many texts such as [Apo, Hi1, Iw2, Kna, Kno, Kob2, La2, Mi2, Ogg1, Ran, Sch, Se2, Shi1]. 1.1. Congruence subgroups Let A = a b c d SL2(Z) act on H, the upper half of the complex plane, by the linear fractional transformation Az = az + b cz + d . The fundamental domain for the action of SL2(Z) on H, which we denote by F, is given by (1.1) F := 1 2 (z) 1 2 and |z| 1 1 2 (z) 0 and |z| = 1 . The end points of the lower boundary of F are the roots of unity i and (1.2) ω := −1 + −3 2 . Modular forms are meromorphic functions which transform in a suitable way with respect to groups of such transformations. For our purposes, we are mostly concerned with certain congruence subgroups of SL2(Z). Definition 1.1. If N is a positive integer, then define the level N congruence subgroups Γ0(N), Γ1(N), and Γ(N) by Γ0(N) := a b c d SL2(Z) : c 0 mod N , Γ1(N) := a b c d SL2(Z) : a d 1 mod N, and c 0 mod N , Γ(N) := a b c d SL2(Z) : a d 1 mod N, and b c 0 mod N . It is straightforward to verify that these groups are indeed subgroups of SL2(Z). 1
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