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 • 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 3 n Central problem in algebraic topology: compute πi(S ) 4 π (Sk) π (Sk) k <k k π>k(S ) 5 6 Values stabilize along diagonals: & &ͮͥ )ͮ& ͍ Ɣ )ͮ&ͮͥ ͍ for ͟ ≫ 0 7 Stable homotopy groups: . & ) ≔ lim )ͮ&ʚ͍ ʛ (finite abelian groups for ͢ Ƙ 0) &→Ϧ Primary decomposition: . ) Ɣ ⨁ ʚ)ʛʚ+ʛ e.g.: ͧ Ɣ Œͦͨ Ɣ Œͬ ⊕ Œͧ + +-$( 8 • Each dot represents a factor of 2, vertical lines indicate additive extensions . e.g.: ʚͧʛʚͦʛ ƔŒͬ, ʚͬʛʚͦʛ ƔŒͦ⨁Œͦ • Vertical arrangement of dots is arbitrary, but meant to suggest patterns 9 10 s (πn )(5) n 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} ͒ ̿∗ʚ͒ʛ • Homotopy invariant: ͚ ≃ ͛ ⇒ ̿ ͚ Ɣ ̿ʚ͛ʛ • Excision: ͔ Ɣ ͒ ∪ ͓ (CW complexes) ⋯ → ̿∗ ͔ → ̿∗ ͒ ⊕ ̿∗ ͓ → ̿∗ ͒ ∩ ͓ → 16 Cohomology theories • Cohomology theories are representable by spectra : – A sequence of pointed spaces so that ) . {̿)} ̿ ͒ = [͒, ̿)] – Consequence of excision: ̿) ≃ Ω̿)ͮͥ • Homotopy groups: ͯ) ) ̿ ≔ )ͮ& ̿& = ̿ (ͤͨ) (Note, in the above, n may be negative) 17 Cohomology theories • Example: singular cohomology – ̿) ͒ = ͂)(͒) – ͂) = ͅ(ℤ, ͢) ℤ, ͢ = 0, – ͂ = Ƥ ) 0, else. • Example: K-theory ͤ Grothendieck group of -vector bundles over – ͅ ͒ = ͅ ͒ = ℂ ͒. , – ͦͅ) = ̼͏ × ℤ ͦͅ)ͮͥ = ͏. ℤ, ͢ even – ͅ = Ƥ ) 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 ) ƣ( 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 ɑ with (̽, ͪ, ͌ ) – ͌ɑ = an ͌-algebra an elliptic curve over – ̽ = ͌′ a non-zero tangent vector at the identity of – ͪ = ̽ an element: ͚ ̽, ͪ ∈ ͌′ such that: ͚ ̽, ͪ = &͚ ̽, ͪ , ∈ (͌ɑ)× 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 ̽c = ℤͮcℤ ∈ ℋ 28 Elliptic Curves and modular forms: a brief review If , a modular form gives a holomorphic function on ͌ ⊆ ℂ ͚ ∈ ͇& ℋ ͚ = ͚(̽c, 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 ę '' 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 ę '' !⊗∗ = ̌ !⊗& &∈ℤ • Define ͎͇̀ ≔ Γę '' • 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 • 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 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 = 8 ƕ 2(ͤ Ǝ 1) Fundamental periods: -periodicity ͪͥ Anomaly at p=2: period = 8 ƕ 2(ͤ Ǝ 1) This anomaly is “explained” by the 8-fold periodicity of KO at the prime 2: 0 0 0 0 0 0 … ∗͉ͅ = ℤ ℤͦ ℤͦ 0 ℤ ℤℤͦ ℤͦ 0 ℤ 46 (π s) Fundamental periods: -periodicity n (5) ͪͦ period = 2(p 2 - 1) = 48 Similarly for : the fundamental -period is ͦ ͤ Ƙ 5 ͪͦ 2(ͤ Ǝ 1) 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+) /{±1} 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 Ǝ 1 ℓ Ǝ 1 ͨ& ͉ͅ = ℤ • Thm(Lipshitz-Sylvester) & Ğ is p-integral, and not p-divisible if (ℓ Ǝ1) & ͤ Ǝ 1 |͟ 56 An analog of J for TMF: NB: is a version of for the congruence subgroup ͎͇̀ͤ ℓ ͎͇̀ Γͤ ℓ < ͍͆ͦ(ℤ) The -theory Hurewicz homomorphism detects much more.