nLab modular form

Contents

Contents

Idea

A modular form is a holomorphic function on the upper half-plane that satisfies certain transformation property under the action of the modular group. Abstractly this transformation property makes the function a section of a certain line bundle on the quotient of the upper half plane that makes it the moduli stack of elliptic curves (over the complex numbers) or more generally a modular curve.

Modular forms are also often called classical automorphic forms, see below. As automorphic forms, they are related to Galois representations as part of the Langlands program.

Modular forms appear as the coefficient ring of the Witten genus on manifolds with rational string structure. For manifolds with actual string structure this refines to topological modular forms, which are the homotopy groups of the spectrum tmf.

Definition

In components

Definition

An (integral) modular form of weight ww is a holomorphic function on the upper half-plane

f:( 2) + f : (\mathbb{R}^2)_+ \hookrightarrow \mathbb{C}

(complex numbers with strictly positive imaginary part)

such that

  1. if A=(a b c d)SL 2()A = \left( \array{a & b \\ c& d}\right) \in SL_2(\mathbb{Z}) acting by A:τ=aτ+bcτ+dA : \tau \mapsto = \frac{a \tau + b }{c \tau + d} we have

    f(A(τ))=(cτ+d) wf(τ) f(A(\tau)) = (c \tau + d)^w f(\tau)

    (notice that for A=(1 1 0 1)A = \left( \array{1 & 1 \\ 0& 1}\right) then f(τ+1)=f(τ)f(\tau + 1) = f(\tau))

  2. ff has at worst a pole at {0}\{0\} (for weak modular forms this condition is relaxed)

    it follows that f=f(q)f = f(q) with q=e 2πiτq = e^{2 \pi i \tau} is a meromorphic funtion on the open disk.

  3. integrality f˜(q)= k=N a kq k\tilde f(q) = \sum_{k = -N}^\infty a_k \cdot q^k then a ka_k \in \mathbb{Z}

More generally there is such a definition for SL(2,)SL(2,\mathbb{Z}) replaced by any other arithmetic subgroup ΓSL(2,)\Gamma \subset SL(2,\mathbb{R}) (e.g. Litt, def.1), giving modular forms on modular curves?.

As functions on lattices

The reference for this definition is Calegari13.

A modular form ff of weight kk over \mathbb{C} is a function on lattices Λ=+τ\Lambda=\mathbb{Z}+\tau\mathbb{Z}\subset \mathbb{C} such that

  1. f(+τ)f(\mathbb{Z}+\tau\mathbb{Z}) is holomorphic as a function of τ\tau

  2. f(μΛ)=μ kf(\mu \Lambda)=\mu^{-k} for all μ ×\mu\in\mathbb{C}^{\times}

  3. f(+τ)f(\mathbb{Z}+\tau\mathbb{Z}) is bounded as τi\tau\to i\infty.

If f(+τ)0f(\mathbb{Z}+\tau\mathbb{Z})\to 0 as τi\tau\to i\infty, we say that ff is a cusp form.

As functions on elliptic curves together with a choice of differential

Again the reference for this definition is Calegari13.

Since elliptic curves over \mathbb{C} can be defined in terms of lattices, the previous definition can also be expressed as follows.

A modular form ff of weight kk over \mathbb{C} is a function on pairs (E,ω)(E,\omega) where EE is an elliptic curve and ωH 0(E,Ω E 1)\omega\in H^{0}(E,\Omega_{E}^{1}), such that

f(E,μω)=μ kf(E,ω)f(E,\mu\omega)=\mu^{-k}f(E,\omega)

and such that f(/+τ)f(\mathbb{C}/\mathbb{Z}+\tau\mathbb{Z}) is bounded as τi\tau\to i\infty.

As sections of a line bundle over the moduli stack

More abstractly, for ell\mathcal{M}_{ell} the moduli stack of elliptic curves (or rather its Deligne-Mumford compactification) and A ellA \to \mathcal{M}_{ell} the corresponding universal bundle, write Ω A/S 1\Omega^1_{A/S} for the line bundle of fiberwise Kähler differential forms. Write ee for the 0-section of this line bundle. Then

ωe *Ω A/S 1 \omega \coloneqq e^\ast \Omega^1_{A/S}

is a line bundle over the moduli stack of elliptic curves. A modular form of weight kk is a section of ω k\omega^{\otimes k}. Using the Kodaira-Spencer isomorphism one can show that the space of cusp forms of weight 22 are the differentials on the corresponding modular curve.

For congruence subgroups

Similarly one considers modular forms for congruence subgroups of the full modular group, hence on the space of elliptic curves with level structure.

As automorphic forms

Instead of regarding, as above, modular forms as sections of a line bundle on a quotient of the upper half plane, one may regard them alternatively as plain functions, but on the (projective) special linear group SL(2,)SL(2,\mathbb{R}). (e.g. Martin 13, section 2, Litt, section 2).

As such these functions are then invariant under the action of the modular subgroup SL(2,)SL(2,)SL(2,\mathbb{Z})\hookrightarrow SL(2,\mathbb{R}) and hence are really functions on the coset space SL(2,)/SL(2,)SL(2,\mathbb{R})/SL(2,\mathbb{Z}) (for forms on moduli of elliptic curves) or more generally Γ\SL(2,)\Gamma \backslash SL(2,\mathbb{Z}) (for forms on more general modular curves?).

This generalizes to the case of other congruence subgroups (as above). Generally such functions on coset spaces like this are called automorphic forms. See there for more.

For the history of the terminology “modular form”/“automorphic form” see also this MO comment.

Properties

Action by Hecke operators

Modular forms can be acted on by Hecke operators (related to Hecke correspondence). Viewing a modular form as a function on lattices, the most common kind of Hecke operator T nT_{n} acts on a modular form ff as follows:

T nf(Λ)= Λf(Λ)T_{n}f(\Lambda)=\sum_{\Lambda'}f(\Lambda')

where the sum runs over all lattices Λ\Lambda' which are index nn subgroups of Λ\Lambda. A modular form which is a simultaneous eigenvector is also called a Hecke eigenform or simply eigenform.

A modular form which vanishes at the cusps (equivalently, whose zeroth Fourier coefficient is equal to 00) is called cuspidal, or a cusp form. A cusp form is normalized if its first Fourier coefficient is equal to 11. For normalized cuspidal eigenforms of integer weight, the eigenvalue of the Hecke operator T nT_{n} is exactly the nn-th Fourier coefficient.

Hecke operators were first developed by Louis Mordell to study the conjectures of Srinivasa Ramanujan on the tau function (the Fourier coefficients of the modular discriminant). They were later developed by (and named after) Erich Hecke who used them to study L-functions.

Analytic continuation of L-functions

Given a cusp form ff of weight kk with Fourier expansion

f(z)= n=1 a ne 2πinzf(z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i n z}

we can associate to it its L-function L(f,s)L(f,s), which is given by

L(f,s)= n=1 a nn sL(f,s)=\sum_{n=1}^{\infty}a_{n}n^{-s}

Alternatively the L-function L(f,s)L(f,s) can be defined via the Mellin transform of ff:

Λ(f,s)=(2π) sΓ(s)L(f,s)= 0 f(iy)y sd ×\Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s)=\int_{0}^{\infty}f(iy)y^{s}d^{\times}

The L-function L(f,s)L(f,s) has an analytic continuation to all of \mathbb{C}.

This is important for instance to formulate conjectures such as the Birch and Swinnerton-Dyer conjecture, which involves the special value of the Hasse-Weil L-function L(E,s)L(E,s) of an elliptic curve at a value of ss where it is originally not defined (the series only converges for s>3/2s \gt 3/2). But the modularity theorem tells us that this Hasse-Weil L-function L(E,s)L(E,s) is equal to the L-function of some cusp form ff, which we know has an analytic continuation to the entire complex plane.

Relation to Galois representations

Modular forms can be used to construct Galois representations. For instance in the case of weight 22 normalized eigenforms this is the Eichler-Shimura construction discussed in DiamondShurman05, chapter 9. This was generalized to higher weight in Deligne71, and the case of weight 11 comes from the work of DeligneSerre74.

The problem of whether a Galois representation comes from some modular form is part of the problem of modularity. Known cases include the modularity theorem of TaylorWiles95 and BreuilConradDiamondTaylor2001. The former is notably an important part of Wiles' proof of Fermat's last theorem.

On a more general level, as modular forms are a special case of automorphic forms, their relation to Galois representations is part of the Langlands program.

Relation of MF 0(2)MF_0(2) to the elliptic genus

Write Γ 0(2)SL 2()\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z}) for the subgroup of the modular group on those elements (a b c d)\left(\array{a & b \\ c & d}\right) for which c=0mod2c = 0\, mod\, 2.

A modular function for Γ 0(2)\Gamma_0(2) is a meromorphic function on the upper half plane which transforms as a modular form under the action of Γ 0(2)SL 2()\Gamma_0(2) \hookrightarrow SL_2(\mathbb{Z}). Write MF (Γ 0(2))MF_\bullet(\Gamma_0(2)) for the ring of these.

There is a natural isomorphism

MF (Γ 0(2))[ϵ,δ] MF_\bullet(\Gamma_0(2)) \simeq \mathbb{C}[\epsilon, \delta]

(see at elliptic genus) for the notation.

(Landweber-Ravenel-Stong 93, theorem 1.5 and sections 5.3, 5.8)

Relation to elliptic cohomology

For EE the elliptic cohomology theory associated to the elliptic curve CC, then

E 2nω(C) n E_{2n} \simeq \omega(C)^{\otimes n}

(where ω\omega is the line bundle from above)

and

E 2n+10. E_{2n+1}\simeq 0 \,.

Examples

References

A basic and handy reference is

  • Pierre Deligne, Courbes elliptiques: formulaire d’apres J. Tate, In Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 53{73. Lecture Notes in Math., Vol. 476. Springer, Berlin, 1975 (web)

Textbook accounts:

Lecture notes and reviews include

The work of Erich Hecke:

  • Erich Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I., Mathematische Annalen 114 (1937) 1–28 [doi:10.1007/BF01594160]

  • Erich Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II., Mathematische Annalen (in German), 114 (1937) 316–351 [doi:10.1007/BF01594180]

The construction of Galois representations associated to normalized eigenforms of weight 2\geq 2 comes from

  • Pierre Deligne, Formes modulaires et représentations l-adiques, Sém. Bourbaki 1968/69, exp. 355, Springer Lecture Notes 179 (1971), 139–172

The construction of Galois representations from weight 11 modular forms comes from

References related to the modularity theorem are

Original discussion in the context of elliptic genera and elliptic cohomology includes

Reviewed in:

Last revised on July 7, 2022 at 13:06:12. See the history of this page for a list of all contributions to it.