2
$\begingroup$

It is well known that a cusp form $$ f = \sum_{n\ge 1}a_n q^n $$ of weight $k$ and level $1$ is determined by its first $d_k = \text{dim } S_k$ coefficients. This follows from the valence formula (which gives a bound) and from an explicit construction using Eisenstein series.

What happens in higher levels, for instance for the group $\Gamma_0(N)$? The valence formula still gives a bound, but can we pinpoint the smallest integer $d_{k,N}$ such that $f \in S_{k}(\Gamma_0(N))$ vanishes if and only if $a_n=0$ for $1\le n \le d_{k,N}$?

What is known and what is conjectured?

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
3
$\begingroup$

In his paper Congruences between Modular Forms, Ram Murty proves a number of results in the spirit of your question. For example, his Theorem 1 is:

Theorem: Let f and g be distinct holomorphic modular forms of weight $k$ and levels $N_1$ and $N_2$. Let $N=\mathrm{Lcm}(N_1,N_2)$. Then for some $$ n \leq \frac{k}{12}N\prod_{p\mid N}\left( 1+\frac{1}{p}\right)$$ we must have $a_f(n)\neq a_g(n)$.

Added: As Kimball has pointed out in a comment, Theorem 4 of this paper is not correct. For more details you should take a look at the paper Distinguishing Hecke Eigenforms by Alexandru Ghitza.

Improve this answer
$\endgroup$
4
  • 1
    $\begingroup$ Though perhaps one should caution the reader that not all of the results in that paper are correct. See Alex Ghitza's "Distinguishing Hecke eigenforms" paper for a corrected version of Murty's Theorem 4. $\endgroup$
    – Kimball
    Commented Jul 8, 2019 at 1:48
  • $\begingroup$ @Kimball - Thanks for pointing that out! I actually wasn't aware that some of the results in the paper weren't correct. I'll edit the post to include a link to Ghitza's paper. $\endgroup$
    – user1073
    Commented Jul 8, 2019 at 1:59
  • $\begingroup$ Sure, I'm not aware of any issues with the other theorems, but I also haven't read that paper. $\endgroup$
    – Kimball
    Commented Jul 8, 2019 at 2:00
  • 1
    $\begingroup$ The above theorem was first proved by Jacob Sturm, "On the congruence of modular forms" (1984). His version has the weaker requirement that $a_f(n)\neq a_g(n)\pmod{M}$ (with a corresponding generalization when the coefficients lie in a number field other than $\mathbb{Q}$). There is some additional discussion of this in Ken Ono's book "The Web of Modularity" in section 2.9. $\endgroup$
    – 2734364041
    Commented Jul 8, 2019 at 3:05

You must log in to answer this question.