Modular forms modulo p

Short description: Mathematical concept

In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms.

## Contents

* 1 Reduction of modular forms modulo 2
* 1.1 Conditions to reduce modulo 2
* 1.2 Basis for modular forms modulo 2
* 1.3 The modular discriminant modulo 2
* 2 Hecke operators modulo 2
* 3 The Hecke algebra modulo 2
* 4 References

## Reduction of modular forms modulo 2

### Conditions to reduce modulo 2

Modular forms are analytic functions, so they admit a Fourier series. As modular forms also satisfy a certain kind of functional equation with respect to the group action of the modular group, this Fourier series may be expressed in terms of q=e2πiz. So if f is a modular form, then there are coefficients c(n) such that f(z)=∑n∈ℕc(n)qn. To reduce modulo 2, consider the subspace of modular forms with coefficients of the q-series being all integers (since complex numbers, in general, may not be reduced modulo 2). It is then possible to reduce all coefficients modulo 2, which will give a modular form modulo 2.

### Basis for modular forms modulo 2

Modular forms are generated by G2 and G3:..[1] It is then possible to normalize G2 and G3 to E2 and E3, having integers coefficients in their q-series. This gives generators for modular forms, which may be reduced modulo 2. Note the Miller basis has some interesting properties [2] Once reduced modulo 2, E2 and E3 are just 1. That is, a trivial reduction. To get a non-trivial reduction, mathematicians use the modular discriminant Δ. It is introduced as a "priority" generator before E2 and E3. Thus, modular forms are seen as polynomials of E2,E3 and Δ (over the complex ℂ in general, but seen over integers ℤ for reduction), once reduced modulo 2, they become just polynomials of Δ over 𝔽2.

### The modular discriminant modulo 2

The modular discriminant is defined by an infinite product:

Δ(q)=q∏n=1∞(1−qn)24=∑n=1∞τ(n)qn.

The coefficients that matches are usually denoted τ, and correspond to the Ramanujan tau function. Results from Kolberg[3] and Jean-Pierre Serre[4] allows to show that modulo 2, we have: Δ(q)≡∑m=0∞q(2m+1)2mod2 i.e., the q-series of Δ modulo 2 consists of q to powers of odd squares.

## Hecke operators modulo 2

Hecke operators are commonly considered as the most important operators acting on modular forms. It is therefore justified to try to reduce them modulo 2.

The Hecke operators for a modular form f are defined as follows[5] Tnf(z)=n2k−1∑a≥1,ad=n,0≤b<dd−2kf(az+bd) with n∈ℕ.

Hecke operators may be defined on the q-series as follows:[5] if f(z)=∑n∈ℤc(n)qn, then Tnf(z)=∑m∈ℤγ(m)qm with

γ(z)=∑a|(n,m),a≥1a2k−1c(mna2).

Since modular forms were reduced using the q-series, it makes sense to use the q-series definition. The sum simplifies a lot for Hecke operators of primes (i.e. when m is prime): there are only two summands. This is very nice for reduction modulo 2, as the formula simplifies a lot. With more than two summands, there would be many cancellations modulo 2, and the legitimacy of the process would be doubtable. Thus, Hecke operators modulo 2 are usually defined only for primes numbers.

With f a modular form modulo 2 with q-representation f(q)=∑n∈ℕc(n)qn, the Hecke operator Tp on f is defined by Tp‾|f(q)=∑n∈ℕγ(n)qn where

γ(n)={c(np) if p∤nc(np)+c(n/p) if p∣n and p an odd prime.

It is important to note that Hecke operators modulo 2 have the interesting property of being nilpotent. Finding their order of nilpotency is a problem solved by Jean-Pierre Serre and Jean-Louis Nicolas in a paper published in 2012:.[6]

## The Hecke algebra modulo 2

The Hecke algebra may also be reduced modulo 2. It is defined to be the algebra generated by Hecke operators modulo 2, over 𝔽2.

Following Serre and Nicolas's notations from[7] ℱ=⟨Δk∣k odd⟩, i.e. ℱ=⟨Δ,Δ3,Δ5,Δ7,Δ9,…⟩. Writing ℱ(n)=⟨Δ,Δ3,Δ5,…,Δ2n−1⟩ so that dim⁡(ℱ(n))=n, define A(n) as the 𝔽2-subalgebra of End(ℱ(n)) given by 𝔽2 and Tp.

That is, if 𝔪(n)={Tp1⋅Tp2⋯Tpk∣p1,p2,…,pk∈ℙ,k≥1} is a sub-vector-space of ℱ, we get A(n)=𝔽2⊕𝔪(n).

Finally, define the Hecke algebra A as follows: Since ℱ(n)⊂ℱ(n+1), one can restrict elements of A(n+1) to ℱ to obtain an element of A(n). When considering the map ϕn:A(n+1)->A(n) as the restriction to ℱ(n), then ϕn is a homomorphism. As A(1) is either identity or zero, A(1)≅𝔽2. Therefore, the following chain is obtained: …->A(n+1)->A(n)->A(n−1)->…->A(2)->A(1)≅𝔽2. Then, define the Hecke algebra A to be the projective limit of the above A(n) as n->∞. Explicitly, this means A=lim<-n∈ℕA(n)={Tp1⋅Tp2⋯Tpk|p1,p2,…,pk∈ℙ,k≥0}.

The main property of the Hecke algebra A is that it is generated by series of T3 and T5.[7] That is: A=𝔽2[Tp∣p∈ℙ]=𝔽2[[T3,T5]].

So for any prime p∈ℙ, it is possible to find coefficients aij(p)∈𝔽2 such that: Tp=∑i+j≥1aij(p)T3iT5j

## References

1. ↑ Stein, William (2007). Modular Forms, a Computational Approach. Graduate Studies in Mathematics. Theorem 2.17. ISBN 978-0-8218-3960-7. https://wstein.org/books/modform/modform/index.html.
2. ↑ Stein, William (2007). Modular Forms, a Computational Approach. Graduate Studies in Mathematics. Lemma 2.20. ISBN 978-0-8218-3960-7. https://wstein.org/books/modform/modform/index.html.
3. ↑ Kolberg, O. (1962). "Congruences for Ramanujan's function τ(n)". Årbok for Universitetet i Bergen Matematisk-naturvitenskapelig Serie (11).
4. ↑ Serre, Jean-Pierre (1973). A course in arithmetic. Springer-Verlag, New York-Heidelberg. p. 96. ISBN 978-1-4684-9884-4.
5. ↑ 5.0 5.1 Serre, Jean-Pierre (1973). A course in arithmetic. Springer-Verlag, New York-Heidelberg. p. 100. ISBN 978-1-4684-9884-4.
6. ↑ Nicolas, Jean-Louis; Serre, Jean-Pierre (2012). "Formes modulaires modulo 2: l'ordre de nilpotence des opérateurs de Hecke". Comptes Rendus Mathématique 350 (7–8): 343–348. doi:10.1016/j.crma.2012.03.013. ISSN 1631-073X. Bibcode: 2012arXiv1204.1036N.
7. ↑ 7.0 7.1 Nicolas, Jean-Louis; Serre, Jean-Pierre (2012). "Formes modulaires modulo 2: structure de l'algèbre de Hecke". Comptes Rendus Mathématique 350 (9–10): 449–454. doi:10.1016/j.crma.2012.03.019. ISSN 1631-073X. Bibcode: 2012arXiv1204.1039N.

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