Lecture 1: Modular Forms and Topology

Lecture 1: Modular forms and Topology

Mark Behrens (MIT) Outline

• Background • Computational – Stable homotopy groups Applications of TMF of spheres – Hurewicz image Cohomology theories – – -self maps – Elliptic curves and – Greek letter elements modular forms • Geometry • What is TMF? – Witten genus – Elliptic cohomology – Derived algebraic – Definition of TMF geometry – Relationship to modular forms

2 Outline

3 n Central problem in algebraic topology: compute πi(S )

4 π (Sk) π (Sk) k k(S )

5 6 Values stabilize along diagonals: for

7 Stable homotopy groups: (finite abelian groups for ) Primary decomposition:

e.g.: 8 • Each dot represents a factor of 2, vertical lines indicate additive extensions e.g.: , • Vertical arrangement of dots is arbitrary, but meant to suggest patterns 9 10 s (πn )(5)

11 Chromatic theory

s • (πk )(p) is built out of chromatic layers

th • The elements of the n layer fit into periodic families ( vn – periodicity )

• Important such families are the “ Greek letter families ” ( α, β, γ...)

• The generic period in the n th chromatic layer is 2(p n-1)

• It is likely no human will know all of the stable homotopy groups of spheres, but it is possible to completely compute a chromatic layer

v1-periodic: completely understood (α – family) v2-periodic: subject of recent work (β – family) v3 and higher: virtually unknown (γ – family and higher)

12 v - periodic layer (π s) 1 n (5) consists solely of α-family

period = 2(p-1) = 8

13 v - periodic layer (π s) 2 n (5) = β−family

period = 2(p 2 - 1) = 48

14 v - periodic layer (π s) 3 n (5) = γ−family

period = 2(p 3 - 1) = 248

15 Cohomology theories

• Use homology/cohomology to study homotopy • A cohomology theory is a contravariant functor

: {Topological spaces} {graded ab groups} ∗ • Homotopy invariant: ≃ ⇒ • Excision: ∪ (CW complexes)

⋯ → ∗ → ∗ ⊕ ∗ → ∗ ∩ →

16 Cohomology theories

• Cohomology theories are representable by spectra : – A sequence of pointed spaces so that . {} = [, ] – Consequence of excision: ≃ Ω

• Homotopy groups:

≔ = ()

(Note, in the above, n may be negative)

17 Cohomology theories

• Example: singular cohomology

– = ()

– = (ℤ, ) ℤ, = 0, – = 0, else. • Example: K-theory Grothendieck group of -vector bundles over – = = ℂ . , – = × ℤ = . ℤ, even – = 0, odd

18 Hurewicz Homomorphism

• A spectrum E is a (commutative) ring spectrum if its associated cohomology theory has “cup products”

∗() is a graded commutative ring

• Such spectra have a Hurewicz homomorphism :

ℎ: ∗ → ∗

Example: detects = ℤ.

19 20 Example: KO (real K-theory) Chern classes and formal groups

• A ring spectrum E is said to be complex orientable if complex vector bundles are orientable in E-cohomology (have a Thom class) • If E is complex orientable, it has Chern classes

• The formal group is the formal power series

defined by the relation on line bundles:

21 Chern classes and formal groups

Example: • = (additive) , = + Example: • = (multiplicative) , =++ (power series expansion of multiplication near 1 in the multiplicative group )

22 Topological modular forms and elliptic cohomology: the rough idea

K-theory: Formal nbhd of 1

Multiplicative group

Elliptic cohomology: Formal nbhd of identity

Elliptic Curve

23 Topological modular forms and elliptic cohomology: the rough idea

• A modular form f associates to each elliptic curve a number

• The cohomology theory of Topological Modular Forms (TMF) consists of the following association: a cohomology class

associates to every elliptic curve C a cohomology class in its associated elliptic cohomology theory:

24 Elliptic Curves and modular forms: a brief review

• An elliptic curve over a ring is a genus 1 curve over (with a marked point)

An elliptic curve over is always of the form • ℂ ℂ/Λ for some lattice . Λ ⊂ ℂ • Elliptic curves are groups (with identity the marked point)

• An elliptic curve has an associated formal group

, ∈ , (obtained by taking power series expansion of multiplication law at the identity)

25 Elliptic Curves and modular forms: a brief review

A modular form (of weight k) over R is a rule which assigns to each tuple with (, , ) – = an -algebra an elliptic curve over – = a non-zero tangent vector at the identity of – = an element: , ∈ such that: , = , , ∈ ()×

Let denote the space of modular forms of weight over

26 Elliptic Curves and modular forms: a brief review

“High-brow perspective”: sections of a line bundle

⊗ ℤ = ℳ ;

27 Elliptic Curves and modular forms: a brief review

“Low-brow perspective”: functions on the upper half-plane Over the complex numbers, every elliptic curve is isomorphic to

ℂ = ℤℤ ∈ ℋ

28 Elliptic Curves and modular forms: a brief review

If , a modular form gives a holomorphic function on ⊆ ℂ ∈ ℋ

= (, 1)

We therefore have:

29 Elliptic Curves and modular forms: a brief review Taking the matrix: we have

Thus f admits a Fourier expansion (q expansion)

We also require a n = 0 for n < 0. (f defined over R => a n ∈ R)

30 Elliptic Curves and modular forms: a brief review

Example: Eisenstein series : E 2k ∈ [M 2k ]Q

Example:

31 Outline

• Background • Computational – Stable homotopy groups Applications of TMF of spheres – Hurewicz image – Cohomology theories – -self maps – Elliptic curves and – Greek letter elements modular forms • Geometry • What is TMF? – Witten genus – Elliptic cohomology – Derived algebraic – Definition of TMF geometry – Relationship to modular forms

32 Elliptic Cohomology theories

Def: An Elliptic spectrum is a tuple

(, , )

Where: • is a commutative ring spectrum • ∗ = [, ], = 2, = . is an elliptic curve over • . • is an isomorphism of formal groups : →

33 Topological Modular Forms

Unfortunately, not every elliptic curve has an associated elliptic cohomology theory. However…

Thm (Goerss-Hopkins-Miller) There exists a sheaf of commutative ring spectra on the etale site of ℳ .

This theorem functorially associates elliptic cohomology theories to elliptic curves which are etale over ℳ . 34 Topological Modular Forms

• Should think of as a topological version of the sheaf ⊗∗ = ⊗ ∈ℤ • Define

≔ Γ

• Analogous to ⊗∗ ∗ ℤ = Γ

TMF is the “mother of all elliptic cohomology theories”

35 Topological Modular Forms

• There is a descent spectral sequence :

⊗ ℳ ; ⇒

• Edge homomorphism:

→ ℤ (rationally this is an iso)

has a bunch of 2 and 3-torsion, and the descent • spectral∗ sequence is highly non-trivial at these primes.

36 The decent spectral sequence for TMF ⊗ 37 (p=2) ℳ ; ⇒ Outline

38 39 Recall: the 2-torsion in real K-theory detects interesting classes in via Hurewicz ∗ 40 Hurewicz image of TMF (p = 2) The decent spectral sequence for TMF Work in progress: (B-Hopkins-Mahowald) (p=2) The complete Hurewicz image: 41 Hurewicz image of TMF (p = 3)

42 Fundamental periods: -periodicity

43 s Fundamental periods: -periodicity (πn )(5)

period = 2(p-1) = 8

Similarly for p > 5: the fundamental -period is 2(p-1) 44 45 Fundamental periods: -periodicity Anomaly at p=2: period = 2( ) Fundamental periods: -periodicity Anomaly at p=2: period = 2( )

This anomaly is “explained” by the 8-fold periodicity of KO at the prime 2:

0 0 0 0 0 0 … ∗ = ℤℤ ℤ 0ℤ ℤℤ ℤ 0ℤ

46 (π s) Fundamental periods: -periodicity n (5)

period = 2(p 2 - 1) = 48

Similarly for : the fundamental -period is 2( ) 47 Fundamental periods: -periodicity

is 144-periodic ∗()

Theorem: (B-Pemmaraju) The fundamental period for -periodic homotopy at the prime 3 is 144.

48 Fundamental periods: -periodicity

49 Fundamental periods: -periodicity

is 192-periodic ∗()

Theorem: (B-Hill-Hopkins-Mahowald) The fundamental period for -periodic homotopy at the prime 2 is 192.

50 51 Fundamental periods: -periodicity J-spectrum and -family

Fix to be a prime which topologically generates ∧ × ℓ (ℤ) ( ∧ × if p = 2) (ℤ) /{±}

Define to be the homotopy fiber

The J-theory Hurewicz homomorphism detects much more.

∗ → ∗

52 53 = detected by KO Hurewicz

= detected by J Hurewicz

J detects all -periodic homotopy s (πn )(5)

= detected by J Hurewicz

54 Greek letter notation: the -family

55 Relationship to Bernoulli numbers

Key points • ℓ acts by multiplication by on ℓ = ℤ • Thm(Lipshitz-Sylvester)

is p-integral, and not p-divisible if (ℓ )

56 An analog of J for TMF:

NB: is a version of for the congruence subgroup ℓ Γ ℓ < (ℤ)

The -theory Hurewicz homomorphism detects much more. (ℓ)

∗ → ∗(ℓ)

57 = detected by Hurewicz

= detected by (ℓ) Hurewicz

58 s 59 (πn )(5)

= detected by (ℓ) Hurewicz 60 s (πn )(5) -torsion in the -family

61 62 Greek Letter Names (Miller-Ravenel-Wilson) -family notation

63 -elements and congruences of modular forms Theorem (B) Let . There is a bijective correspondence:

64 Outline

65 Geometry of TMF: survey

• Question: What is the geometric nature of TMF? • E.g. K-theory cocycles are given by vector bundles, what gives a TMF-cocycle? • Beginning with Witten and Segal, and elaborated on by Stolz-Teichner, et. al., the belief is that a TMF- cocycle is given by a “ conformal field theory ”. Much is conjectural. • Lurie shows that TMF has an algebro-geometric significance, as the “derived” moduli space of elliptic curves .

66 Genera

Let be a suitable group over , andlet

manifoldswithstablenormalstructure Ω = cobordism

An -valued genus is a graded ring homomorphism ∗

Φ: Ω∗ → ∗

67 Examples of genera These all arise from maps of commutative ring spectra • Cardinality of 0-manifolds (mod 2) Ω∗ → ℤ → ℤ

• Signed cardinality of oriented 0-manifolds Ω∗ → ℤ → ℤ The -genus • Ω∗ → ∗ → The -genus of a spin manifold is the index of the Dirac operator acting on the sections of the associated spinor bundle

68 Witten Genus

Witten produced a genus W: Ω∗ → ∗ ℤ

( 7-connected cover of = )

The idea: a string structure is a vanishing of the obstruction to quantizing a supersymmetric conformal field theory on a manifold. The partition function of the resulting QFT associates a number to every elliptic curve – a modular form! Kevin Costello has a renormalization framework that actually makes some version of this statement mathematically precise 69 Witten Genus

Witten produced a genus W: Ω∗ → ∗ ℤ

( 7-connected cover of = )

Theorem(Ando-Hopkins-Rezk) The Witten genus refines to a map of ring spectra : →

70 “Hierarchy of genera”

71 Derived Algebraic Geometry (Lurie’s approach)

A derived scheme consists of • An ordinary scheme , • A sheaf of commutative ring spectra such that

= with a certain additional local condition…

A derived elliptic curve is a derived abelian group scheme whose underlying scheme is an elliptic curve.

72 Derived Algebraic Geometry (Lurie’s approach)

Let be a ring spectrum. An orientation of a derived elliptic curve is an isomorphism / Spf(ℂ ) → ≃

Theorem(Lurie) The moduli problem of oriented derived elliptic curves is representable. The representing Deligne-Mumford stack is

(ℳ , )

73 Advantages to the DAG approach

• Gives a “ pure thought ” construction of TMF – Goerss-Hopkins-Miller rely on obstruction theory • Gives a homotopically unique construction of TMF – the moduli space of solutions to the Goerss-Hopkins- Miller obstruction problem is not contractible (but does have one component). • Generalizes to give equivariant TMF for compact Lie groups, a “genuine” equivariant theory in the sense of Lewis-May-Steinberger

74 Objectives for next 2 lectures:

Chromatic Group Arithmetic Cohomology Cocycles Geometry level Object theory represeted by: multiplicative Vector 1 GL 1 K-theory Spin group bundles

elliptic conformal 2 GL 2 TMF String curves field theories? n ? ? ? ? ?

How does this generalize for arbitrary n?

75 Objectives for next 2 lectures:

Chromatic Group Arithmetic Cohomology Cocycles Geometry level Object theory represeted by: multiplicative Vector 1 GL 1 K-theory Spin group bundles

elliptic curves conformal 2 GL 2 TMF String field theories? n U(1,n-1) abelian TAF ?? varieties with complex multiplication