MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The space $M_k(1)$ of modular forms of level $1$ has a unique basis $\left\{ f_0,f_1,\ldots,f_d\right\}$ such that $a_i(f_j) = \delta_{ij}$ for all $0\leq i,j \leq d$, and $f_i \in \mathbb{Z}[[q]]$ for all $0 \leq i \leq d$. This is often called the Miller basis for $M_k(1)$.

Does there exist a basis in $\mathbb{Z}[[q]]$ for $M_k(N)$?

This would seem quite incredible. Maybe the following analogon of the Miller basis might be true in higher level. Let us denote $M_k(N,\mathbb{Z})$ the elements in $M_k(N) \cap \mathbb{Z}[[q]]$.

Is $M_k(N,\mathbb{Z})$ a free $\mathbb{Z}$-module? If so, what is its rank in function of $N,k$?

share|cite|improve this question
up vote 3 down vote accepted

The space $M_k(N)$ has a basis in $\mathbb{Z}[[q]]$ for any $N$ and $k$. This is a straightforward consequence of Eichler-Shimura theory and will be in any decent textbook on modular forms (e.g. Diamond + Shurman, Miyake, Lang).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.