nLab modular form

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 $w$ is a holomorphic function on the upper half-plane

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

(complex numbers with strictly positive imaginary part)

such that

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

   $$f(A(\tau)) = (c \tau + d)^w f(\tau)$$

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

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

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

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

More generally there is such a definition for $SL(2,\mathbb{Z})$ replaced by any other arithmetic subgroup $\Gamma \subset SL(2,\mathbb{R})$ (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 $f$ of weight $k$ over $\mathbb{C}$ is a function on lattices $\Lambda=\mathbb{Z}+\tau\mathbb{Z}\subset \mathbb{C}$ such that

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

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

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

If $f(\mathbb{Z}+\tau\mathbb{Z})\to 0$ as $\tau\to i\infty$, we say that $f$ 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 $f$ of weight $k$ over $\mathbb{C}$ is a function on pairs $(E,\omega)$ where $E$ is an elliptic curve and $\omega\in H^{0}(E,\Omega_{E}^{1})$, such that

$$f(E,\mu\omega)=\mu^{-k}f(E,\omega)$$

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

As sections of a line bundle over the moduli stack

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

$$\omega \coloneqq e^\ast \Omega^1_{A/S}$$

is a line bundle over the moduli stack of elliptic curves. A modular form of weight $k$ is a section of $\omega^{\otimes k}$. Using the Kodaira-Spencer isomorphism one can show that the space of cusp forms of weight $2$ 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 $SL(2,\mathbb{R})$. (e.g. Martin 13, section 2, Litt, section 2).

As such these functions are then invariant under the action of the modular subgroup $SL(2,\mathbb{Z})\hookrightarrow SL(2,\mathbb{R})$ and hence are really functions on the coset space $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ (for forms on moduli of elliptic curves) or more generally $\Gamma \backslash SL(2,\mathbb{Z})$ (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 $T_{n}$ acts on a modular form $f$ as follows:

$$T_{n}f(\Lambda)=\sum_{\Lambda'}f(\Lambda')$$

where the sum runs over all lattices $\Lambda'$ which are index $n$ subgroups of $\Lambda$. 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 $0$) is called cuspidal, or a cusp form. A cusp form is normalized if its first Fourier coefficient is equal to $1$. For normalized cuspidal eigenforms of integer weight, the eigenvalue of the Hecke operator $T_{n}$ is exactly the $n$-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 $f$ of weight $k$ with Fourier expansion

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

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

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

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

$$\Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s)=\int_{0}^{\infty}f(iy)y^{s}d^{\times}$$

The L-function $L(f,s)$ 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 $L(E,s)$ of an elliptic curve at a value of $s$ where it is originally not defined (the series only converges for $s \gt 3/2$). But the modularity theorem tells us that this Hasse-Weil L-function $L(E,s)$ is equal to the L-function of some cusp form $f$, 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 $2$ 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 $1$ 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 $MF_0(2)$ to the elliptic genus

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

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

There is a natural isomorphism

$$MF_\bullet(\Gamma_0(2)) \simeq \mathbb{C}[\epsilon, \delta]$$

(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 $E$ the elliptic cohomology theory associated to the elliptic curve $C$, then

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

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

and

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

Examples

• Eisenstein series

• j-invariant

• Weierstrass sigma-function

• Dedekind eta function

Related concepts

• modular group

• elliptic curve, moduli stack of elliptic curves

• Hecke correspondence

• Jacobi form

• elliptic genus, Witten genus

• topological modular form, tmf

• p-adic modular form

• generalization to functions on moduli of higher dimensional abelian varieties: Hilbert modular form, Siegel modular forms

• modularity theorem

• Rankin-Cohen bracket

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:

• Jean-Pierre Serre, chapter VII of: A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer (1973) [doi:10.1007/978-1-4684-9884-4, pdf]

• N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Math. 97 Springer (1984, 1993)

• Friedrich Hirzebruch, Thomas Berger, Rainer Jung, Appendix I of: Manifolds and Modular Forms, Aspects of Mathematics 20, Viehweg (1992), Springer (1994) [doi:10.1007/978-3-663-10726-2, pdf]

• Fred Diamond, Jerry Shurman, A First Course in Modular Forms, GTM 228, Springer (2005) [doi:10.1007/978-0-387-27226-9, ISBN-13: 978-0387232294]

• William Stein: Modular Forms, a Computational Approach, Graduate Studies in Mathematics 79, AMS (2007) [doi:10.1090/gsm/079]

• Max Koecher, Aloys Krieg: Elliptische Funktionen und Modulformen, Springer (2007) [doi:10.1007/978-3-540-49325-9]

Lecture notes and reviews:

• Richard Hain, section 4 of: Lectures on Moduli Spaces of Elliptic Curves [arXiv:0812.1803]

• Charles Rezk, section 10 of pdf

• Daniel Litt, Automorphic forms notes, part I (pdf)

• Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, Don Zagier, The 1-2-3 of modular forms, Lectures at a Summer School 2004 in Nordfjordeid, Norway; Universitext, Springer 2008.

• Nora Ganter, Topological modular forms literature list

• Kimball Martin, A brief overview of modular and automorphic forms, 2013 pdf

• Frank Calegari, Congruences between modular forms, 2013 pdf

• Joachim Wehler: Modular Forms and Elliptic Curves (2021) [pdf]

• Wikipedia, Modular form

Introduction in relation to string theory:

• Eric D'Hoker, Justin Kaidi, Lectures on modular forms and strings [arXiv:2208.07242]

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 $\geq 2$ 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 $1$ modular forms comes from

• Pierre Deligne and Jean-Pierre Serre Formes modulaires de poids 1 Ann. Sci. Ecole Norm. Sup. (4), 7:507–530 (1975), 1974.

References related to the modularity theorem are

• Richard Taylor, Andrew Wiles, Ring-Theoretic Properties of Certain Hecke Algebras, Annals of Mathematics Second Series 141 3 (1995) 553-572 $[$doi:10.2307/2118560$]$

• Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor, On the modularity of elliptic curves over $\mathbf{Q}$: Wild 3-adic exercises, Journal of the American Mathematical Society 14 4 (2001) 843–939 $[$doi:10.1090/S0894-0347-01-00370-8$]$

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

• Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)

• Peter Landweber, Elliptic Cohomology and Modular Forms, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326 (1988) 55-68 [doi:10.1007/BFb0078038]

Reviewed in:

• Kefeng Liu, Modular forms and topology, 1996 (citeseer)