Dimension Formulas¶

When computing with spaces of modular forms, it is helpful to have easy-to-compute formulas for dimensions of these spaces. Such formulas provide a check on the output of the algorithms from Chapter General Modular Symbols that compute explicit bases for spaces of modular forms. We can also use dimension formulas to improve the efficiency of some of the algorithms in Chapter General Modular Symbols, since we can use them to determine the ranks of certain matrices without having to explicitly compute those matrices. Dimension formulas can also be used in generating bases of -expansions; if we know the dimension of and if we have a process for computing -expansions of elements of , e.g., multiplying together -expansions of certain forms of smaller weight, then we can tell when we are done generating .

This chapter contains formulas for dimensions of spaces of modular forms, along with some remarks about how to evaluate these formulas. In some cases we give dimension formulas for spaces that we will define in later chapters. We also give many examples, some of which were computed using the modular symbols algorithms from Chapter General Modular Symbols.

Many of the dimension formulas and algorithms we give below grew out of Shimura’s book [Shi94] and a program that Bruce Kaskel wrote (around 1996) in PARI, which Kevin Buzzard extended. That program codified dimension formulas that Buzzard and Kaskel found or extracted from the literature (mainly [Shi94, Section 2.6]). The algorithms for dimensions of spaces with nontrivial character are from [CO77], with some refinements suggested by Kevin Buzzard.

For the rest of this chapter, denotes a positive integer and is an integer. We will give no simple formulas for dimensions of spaces of weight modular forms; in fact, it might not be possible to give such formulas since the methods used to derive the formulas below do not apply in the case . If , the only modular forms are the constants, and for the dimension of is .

For a nonzero integer and a prime , let be the largest integer such that . In the formulas in this chapter, always denotes a prime number. Let be the space of modular forms of level weight and character , and let and be the cuspidal and Eisenstein subspaces, respectively.

The dimension formulas below for , , and can be found in [DS05, Ch. 3], [Shi94, Section 2.6] [1] and [Miy89, Section 2.5]. They are derived using the Riemann-Roch Theorem applied to the covering or and appropriately chosen divisors. It would be natural to give a sample argument along these lines at this point, but we will not since it easy to find such arguments in other books and survey papers (see, e.g., [DI95]). So you will not learn much about how to derive dimension formulas from this chapter. What you will learn is precisely what the dimension formulas are, which is something that is often hard to extract from obscure references.

In addition to reading this chapter, the reader may wish to consult [Mar05] for proofs of similar dimension formulas, asymptotic results, and a nonrecursive formula for dimensions of certain new subspaces.

Modular Forms for ¶

For any prime and any positive integer , let be the power of that divides . Also, let

Note that is the index of in (see Exercise 6.1).

Proposition 6.1

We have , and for even,

The dimension of the Eisenstein subspace is

The following is a table of for some values of and :

Example 6.2

Use the commands dimension_cusp_forms, dimension_eis, and dimension_modular_forms to compute the dimensions of the three spaces , and , respectively. For example,

sage: dimension_cusp_forms(Gamma0(2007),2)
221
sage: dimension_eis(Gamma0(2007),2)
7
sage: dimension_modular_forms(Gamma0(2007),2)
228


Remark 6.3

Csirik, Wetherell, and Zieve prove in [CWZ01] that a random positive integer has probability of being a value of

and they give bounds on the size of the set of values of below some given . For example, they show that are the first few integers that are not of the form for any . See Figure 6.1 for a plot of the very erratic function . In contrast, the function is very well behaved (see Figure 6.2).

Figure 6.1

Dimension of as a function of

Figure 6.2

Dimension of as a function of .

New and Old Subspaces¶

In this section we assume the reader is either familiar with newforms or has read Section Atkin-Lehner-Li Theory.

For any integer , let

where the product is over primes that exactly divide . Note that is not the Moebius function, but it has a similar flavor.

Proposition 6.4

The dimension of the new subspace is

where the sum is over the positive divisors of . As a consequence of Theorem 9.4, we also have

where is the number of divisors of .

Example 6.5

We compute the dimension of the new subspace of using the Sage command dimension_new_cusp_forms as follows:

sage: dimension_new_cusp_forms(Gamma0(11),12)
8
sage: dimension_cusp_forms(Gamma0(11),12)
10
sage: dimension_new_cusp_forms(Gamma0(2007),12)
1017
sage: dimension_cusp_forms(Gamma0(2007),12)
2460


Modular Forms for ¶

This section follows Section Modular Forms for closely, but with suitable modifications with replaced by .

Define functions of a positive integer by the following formulas:

Note that is the genus of the modular curve (associated to ) and is the number of cusps of .

Proposition 6.6

We have . If , then so

where is given by the formula of Proposition 6.1. If , let

Then for ,

The dimension of the Eisenstein subspace is as follows:

The dimension of the new subspace of is

where is as in the statement of Proposition 6.4.

Remark 6.7

Since , the formulas above for and also yield a formula for the dimension of .

Figure 6.3

Dimension of as a function of .

The following table contains the dimension of for some sample values of and :

Example 6.8

We compute dimensions of spaces of modular forms for :

sage: dimension_cusp_forms(Gamma1(2007),2)
147409
sage: dimension_eis(Gamma1(2007),2)
3551
sage: dimension_modular_forms(Gamma1(2007),2)
150960


Modular Forms with Character¶

Fix a Dirichlet character of modulus , and let be the conductor of (we do not assume that is primitive). Assume that , since otherwise and the formulas of Section Modular Forms for apply. Also, assume that , since otherwise . In this section we discuss formulas for computing each of , and .

In [CO77], Cohen and Oesterl’e assert (without published proof; see Remark Remark 6.11 below) that for any and , as above,

where is as in Section Modular Forms for , and , and are

It remains to define . Fix a prime divisor and let . Then

This flexible formula can be used to compute the dimension of , , and for any , , , by using that

One thing that is not straightforward when implementing an algorithm to compute the above dimension formulas is how to efficiently compute the sets and . Kevin Buzzard suggested the following two algorithms. Note that if is odd, then , so the sum over is only needed when is even.

Algorithm 6.9

Given a positive integer and an even Dirichlet character of modulus , this algorithm computes .

1. [Factor ] Compute the prime factorization of .

2. [Initialize] Set and .

3. [Loop Over Prime Divisors] Set . If , return . Otherwise set and .

1. If , return .
2. If and , return .
3. If and , go to step (3).
4. Compute a generator using Algorithm 4.4.
5. Compute .
6. Use the Chinese Remainder Theorem to find such that and .
7. Set .
8. Set .
1. If , set and go to step (3).
1. If , set and go to step (3).

Proof

Note that , since is even. By the Chinese Remainder Theorem, the set is empty if and only if there is no square root of modulo some prime power divisor of . If is empty, the algorithm correctly detects this fact in steps (a)(b). Thus assume is nonempty. For each prime power that exactly divides , let be such that and for . This is the value of computed in steps (d)(g) (as one sees using elementary number theory).

The next key observation is that

(1)

since by the Chinese Remainder Theorem the elements of are in bijection with the choices for a square root of modulo each prime power divisors of . The observation (1) is a huge gain from an efficiency point of view—if had prime factors, then would have size , which could be prohibitive, where the product involves only factors. To finish the proof, just note that steps (h)(j) compute the local factors , where again we use that is even. Note that a solution of lifts uniquely to a solution mod for any , because the kernel of the natural homomorphism is a group of -power order.

The algorithm for computing the sum over is similar.

For , to compute , use the formula directly and the fact that , unless and . To compute for , use the fact that the big formula at the beginning of this section is valid for any integer to replace by and that for to rewrite the formula as

Note also that for , if and only if is trivial and it equals otherwise. We then also obtain

We can also compute when directly, since

The following table contains the dimension of for some sample values of and . In each case, is the product of characters of maximal order corresponding to the prime power factors of (i.e., the product of the generators of the group of Dirichlet characters of modulus ).

Example 6.10

We compute the last line of the above table. First we create the character .

sage: G = DirichletGroup(2007)
sage: e = prod(G.gens(), G(1))


Next we compute the dimension of the four spaces.

sage: dimension_cusp_forms(e,2)
222
sage: dimension_cusp_forms(e,3)
0
sage: dimension_cusp_forms(e,4)
670
sage: dimension_cusp_forms(e,24)
5150


We can also compute dimensions of the corresponding spaces of Eisenstein series.

sage: dimension_eis(e,2)
4
sage: dimension_eis(e,3)
0
sage: dimension_eis(e,4)
4
sage: dimension_eis(e,24)
4


Remark 6.11

Cohen and Oesterl’e also give dimension formulas for spaces of half-integral weight modular forms, which we do not give in this chapter. Note that [CO77] does not contain any proofs that their claimed formulas are correct, but instead they say only that “Les formules qui les donnent sont connues de beaucoup de gens et il existe plusieurs m’ethodes permettant de les obtenir (th’eoreme de Riemann-Roch, application des formules de trace donn’ees par Shimura).” [2] Fortunately, in [Que06], Jordi Quer derives the (integral weight) formulas of [CO77] along with formulas for dimensions of spaces and for more general congruence subgroups.

Let be the conductor of a Dirichlet character of modulus . Then the dimension of the new subspace of is

where is as in the statement of Proposition 6.4, and is the restriction of mod .

Example 6.12

We compute the dimension of for a quadratic character of modulus .

sage: G = DirichletGroup(2007, QQ)
sage: e = prod(G.gens(), G(1))
sage: dimension_new_cusp_forms(e,2)
76
`

Exercises¶

Exercise 6.1

Let and be as in this chapter.

1. Prove that .
2. Prove that for , , so is the index of in .

Exercise 6.2

Use Proposition 6.4 to find a formula for . Verify that this formula is the same as the one in Corollary Corollary 2.16.

Exercise 6.3

Suppose either that or that is prime and . Prove that .

Exercise 6.4

Fill in the details of the proof of Algorithm 6.9.

Exercise 6.5

Implement a computer program to compute as a function of and .

Footnotes

 [1] The formulas in [Shi94, Section 2.6] contain some minor mistakes.
 [2] The formulas that we give here are well known and there exist many methods to prove them, e.g., the Riemann-Roch theorem and applications of the trace formula of Shimura.

Previous topic

Eisenstein Series and Bernoulli Numbers

Linear Algebra