MODULAR FORMS-page77

# MODULAR FORMS-page77 - 73 Proof Let be a normal subgroup of...

This preview shows page 1. Sign up to view the full content.

73 Proof. Let Γ be a normal subgroup of finite index in Γ(1) which is contained in Γ. It always can be found by taking the intersection of conjugate subgroups g - 1 · Γ · g, g Γ(1). We first apply Lemma 7.3 to the case when B = M (Γ ) , A = M (Γ(1)). Since A = C [ T 1 , T 2 ] is finitely generated, B is finitely generated. It follows easily from (7.15) that the field of fractions of B is a finite extension of the field of fractions of A of degree equal to the order of the group Γ / Γ . Next we apply the same lemma to the case when B = M (Γ ) , A = M (Γ) . Then B is finitely generated, hence A is finitely generated. Corollary 7.3. The linear spaces M k (Γ) are finite-dimensional. Proof. Let f 1 , . . . , f k be a set of generators of the algebra M k (Γ). Writing each f i as a linear combination of modular forms of different weights, and then adding to the set of generators all the summands, we may assume that M k (Γ) is generated by finitely many modular forms f i ∈ M k i (Γ) , i = 1 , . . . , n . Now M k (Γ) is spanned as a vector space over C
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern