Elliptic modular forms Hironori Shiga Version May 10, 2008 Contents 1 SL2(Z) and elliptic curves 2 1.1 SL2(Z) and the moduli of complex tori . 2 1.2 The Fundamental region and a system of generators . 3 1.3 The Weierstrass } function . 5 1.4 Nonsingular cubics and the invariant j ............................. 9 1.5 Elliptic modular function j(¿).................................. 11 2 Modular forms for SL2(Z) 14 2.1 Cusps . 14 2.2 Concept of modular forms . 15 2.3 Eisenstein series . 16 2.4 Discriminant form . 19 2.5 Eisenstein series E2(z)...................................... 19 2.6 Algebra M(¡)........................................... 21 2.7 The Dedekind ´ function . 24 3 Modular form for congruence subgroups 26 3.1 Geometry of congruence subgroups . 27 3.2 Principal congruence subgroup ¡(N).............................. 27 3.3 Recalling the Riemann-Roch theorem . 30 3.4 Dimension formula for congruence subgroups . 30 4 Hecke operators and Hecke eigen forms 34 4.1 Preparatory consideration . 34 4.2 Hecke operator T (n)....................................... 35 4.3 Hecke eigen form . 37 4.4 Examples . 39 5 Theta functions 39 6 An example: how it works the theory applied to number theoretic problems 40 1 1 SL2(Z) and elliptic curves 1.1 SL2(Z) and the moduli of complex tori De¯nition 1.1. For !1;!2 2 C ¡ f0g with ¿ = !2=!1 2= R, we de¯ne a lattice in C by ¤ = ¤(!1;!2) = fm!1 + n!2 : m; n 2 Zg: Proposition 1.1. We have 0 0 () 9 2 0 0 ¤(!1;!2) = ¤(!1;!2) M GL(2; Z); (!1;!2) = (!1;!2)M: Proposition 1.2. For ¿; ¿ 0 2 H = fz 2 C : Im (z) > 0g , we have ( ¶ a b a¿ + b ¤(1;¿ 0) = k¤(1;¿); (k 2 C ¡ f0g) () 9M = 2 SL (Z); ¿ 0 = : c d 2 c¿ + d De¯nition 1.2. We de¯ne a complex torus T (!1;!2) = C=¤(!1;!2);T (¿) = C=¤(1;¿): Theorem 1.1. ( ¶ a b a¿ + b T (¿) » T (¿ 0) () 9M = 2 SL (Z); ¿ 0 = : biholo c d 2 c¿ + d [proof]. ((=) is apparaent from the above Proposition. We show (=)). Let ¼(resp. ¼0) be the canonical projection from C to T (¿)( resp.T (¿ 0)). And let f : T (¿) ! T (¿ 0) be a biholomorphic map. We suppose f(O) = O0, with ¼(0) = O; ¼0(0) = O0. f 0 ~ has a lifting f1 : C ! T (¿ ) in a natural way. It induces a analytic function f : U ! C de¯ned in a neighborhood U of 0. And f~ has an analytic continuation on the whole plane C as a possibly multivalued analytic function. Here, by the Monodromy theorem f~ is single valued. f~ C - C ¼ ¼0 ? f ? T (¿) - T (¿ 0) Diagram 1：Lifting f~ of f If we consider f~¡1, by the Monodromy theorem again, we can show f~ is injective. So f~ : C ! f~(C) is a biholomorphic map. By the Riemann uniformization theorem, the image f~(C) cannot be a proper subdomain. So f~ : C ! C is a bijective map, and it is biholomorphic. By the Weierstrass singularity theorem we can show that Aut(H) is a group of nontrivial linear functions. Hence we may put f~(z) = kz; note that we supposed f(O) = O0. So we have ¤(1;¿ 0) = k¤(1;¿). q.e.d. 2 1.2 The Fundamental region and a system of generators Notations: P SL2(R) = SL2(R)= § I = Aut(H); ¡ := SL2(Z); ¡ := SL2(Z)= § I; ½ ¡ \ in general,( for a¶ subgroup G ¡; G := G=(< I > G); a b ¡(N) := f 2 ¡) : a ´ d ´ 1 (mod N ); b ´ c ´ 0 (mod N )g C ¡(N 2 Z+): c d De¯nition 1.3. ¡(N) is called the principal congruence subgroup of level N. Remark 1.1. Note ¡I2 = ¡(N) N > 2 . So ¡(2) = ¡(2)= § I; ¡(N) = ¡(N)(N > 2): De¯nition 1.4. ( ¶ a b ¡ (N) := f 2 ¡: c ´ 0 (mod N )g; 0 c d ( ¶ a b ¡ (N) := f 2 ¡ (N): a ´ d ´ 1 (mod N )g: 1 c d 0 De¯nition 1.5. Let G be a subgroup of ¡. G acts on H. Two points z1 and z2 are said to be G-equivalent, if we have z2 = g(z1) for some g 2 G. A closed region in H is said to be a fundamental region of G, if (1) Every point z 2 H is G-equivalent to a point in F . (2) Any two di®erent points in the interior of F are not G-equivalent. Theorem 1.2. The closed region 1 F = fz 2 H : jRe zj · and jzj ¸ 1g (1.1) 2 is a fundamental region for ¡. F i e2 Πi3 eΠi3 0 Re - 1 1 2 2 Fundamental region for ¡ [proof]. (i) F contains all representatives. Set ( ¶ ( ¶ 1 1 0 ¡1 T = ;S = : 0 1 1 0 3 ( ¶ a b G := hS; T i ½ ¡. For a ¯xed z 2 H and g = 2 G we have c d Im z Im g(z) = : jcz + dj2 2 2 fjcd + dj : c; d 2 Z ¡ f(0; 0)g; coprimeg has the minimum, so Im g(z) has the maximum. Let g0 realizes k § ¢ ¢ ¢ k0 2 j j · 1 the maximum, so do T g0 (k = 0; 1; ). And we can ¯nd g = T g0 G such that g(z) 2 . We have g(z) 2 F . In fact, if we have g(z) 2= F , it holds jg(z)j < 1. Take S(g(z)). We have Im g(z) Im S(g(z)) = > Im g(z): jg(z)j2 This is contradicts the maximality. So we could ¯nd a g 2 G such that g(z) 2 F . (ii) Equivalent points. ( ¶ a b Suppose z1; z2 are equivalent, and we have z2 = g(z1); g = 2 ¡ ¡ f§Ig. Assume Im z2 = p c d Im z1 ¸ j j ¢ 3 · j j · j j j j · 2 Im z1 . So we have c c Im z1 (c Re z1 + d) + ic Im z1 = cz1 + d 1. We can jcz1+dj 2 see that it must hold jcj · 1. By observing jcz1j · 1=2 we have jdj · 1. So we have only restricted possibilities: (a) c = 0; d = §1 =) g : az + b § ) ¡ 1 (b) c = 1; d = 0 = g : a z § ) ¡1 2¼i=3 (c) c = d = 1 = g : a + z+1 and z1 = e ¡ § ) ¡1 ¼i=3 (d) c = d = 1 = g : a + z¡1 and z1 = e . In any case z1 and z2 cannot stay inside F at the same time. -1 -1 F2 F z + 1 z - 1 1 i i e2 Πi3 eΠi3 ¬ ® - 1 1 - 1 1 2 0 2 2 0 2 Figures of case (c)(d) q.e.d. Proposition 1.3. Let z1; z2 2 @F . z1 and z2 are ¡-equivalent () (1) z2 ¡ z1 = §1; jRe z1j = jRe z2j = 1=2 or 1 (2) z2 = ¡ ; jz1j = jz2j = 1. z1 We use the notation Gz = fg 2 G : g(z) = zg, the isotropy group for z. Set ( ¶ ( ¶ 1 1 0 ¡1 T = ;S = : 0 1 1 0 4 Proposition 1.4. We have (i) ¡i = f§I; §Sg 2 2¼i=3 (ii) ¡! = f§I; §ST; §(ST ) g;! = e 2 2¼i=6 (iii) ¡¡! = f§I; §T S; §(TS) g; ¡! = e (iv) ¡z = §I; otherwise: [proof]. We can solve cz2 + (d ¡ a)z ¡ b = 0; ad ¡ bc = 1 (a; b; c; d 2 Z) for z = i; !; ¡!2. Corollary 1.1. ¡ acts on H as a discrete group. Theorem 1.3. We have ¡ = hS; T: § idi: [proof]. Take an element γ 2 ¡. According to the part (i) of the proof of Theorem 1.2 we can ¯nd g 2 G = hS; T i so that we have gγ(2i) 2 F . Namely it holds gγ(2i) = 2i. So gγ 2 I2i = f§idg. Hence we obtain the required conclusion. q.e.d. 1.3 The Weierstrass } function De¯nition 1.6. Let ¤ = ¤(!1;!2) be a lattice in C. The Weierstrass } function is de¯ned by ( ¶ 1 X 1 1 }(z) = + ¡ : (1.2) z2 (z ¡ !)2 !2 !2¤¡f0g Remark 1.2. For a lattice ¤, 0 0 X 1 X X ( means the sum ) !k !2¤¡f0g is absolute convergent for k ¸ 3, and is conditional convergent for k = 2. Theorem 1.4. (i) }(z) is meromorphic on C and doubly periodic, i.e. }(z + !) = }(z)(! 2 ¤): (ii) }(z) has double poles at z = ! 2 ¤, and is an even function of order 2. 0 (iii) } is an odd function of order 3. It has zeros at half lattice points z = !1=2;!2=2; (!1 + !2)=2, and has a triple poles at z = ! 2 ¤. 0 0 [proof]. (i) We have } (z + !i) = } (z)(i = 1; 2). So }(z + !i) ¡ }(z) = ci (c : constant). By putting z = ¡!i=2 we get ci = 0. (ii)(iii) follows from the following general argument. Lemma 1.1. (i) Let f be a doubly periodic meromorphic function (=6 0) for a lattice ¤. f takes every complex value ® same times (counting multiplicities) in a period parallelogram Pa = f¸!1 + ¹!2 + a : 0 < ¸ < 1; 0 < ¹ < 1g provided f =6 ®; 1 on @Pa. 5 (ii) Let f be a doubly periodic meromorphic function. a1; : : : ; ar be its representatives of zeros, and b1; : : : ; br be the representatives of poles. Then we have a1 + ::: + ar ¡ (b1 + ::: + br) 2 ¤: [proof]. (i) By the argument principle Z 0 1 f f 2 g ¡ f 2 1g ¡ dz = ] z Pa : f(z) = ® ] z Pa : f(z) = : 2¼i @Pa f ® By the periodicity the left hand side is equal to 0.