The etale theta function of S. Mochizuki Kiran S. Kedlaya - PowerPoint PPT Presentation

The ´ etale theta function of S. Mochizuki Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ The IUT theory of Shinichi Mochizuki Clay Mathematics Institute Oxford University, December 9-10, 2015 Reference: [EtTh] S. Mochizuki, The ´ etale theta function and its frobenioid-theoretic manifestations, Publ. RIMS 45 (2009), 227–349. Disclaimers: Some notation differs from [EtTh]. All mistakes herein are due to the speaker’s misunderstandings of [EtTh]; moreover, no true understanding of IUT should be inferred. Feedback is welcome! Supported by NSF (grant DMS-1501214), UCSD (Warschawski chair), Guggenheim Fellowship. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 1 / 37

Motivation: review of Bogomolov-Zhang Setup (after Zhang’s lecture) Let C be a compact Riemann surface of genus g . Let S be a finite subset of C . Let f : E → C be a nonisotrivial elliptic fibration over C − S with stable reduction at all s ∈ S . Let ∆ E = � s ∆ s · ( s ) be the discriminant divisor; we will prove 1 deg ∆ E ≤ 6(# S + 2 g − 1) . To begin, fix a point p ∈ C − S ; we then have a monodromy action ρ : π 1 ( C − S , p ) → SL( H 1 ( E p , Z )) ∼ = SL 2 ( Z ) . The group π 1 ( C − S , p ) is generated by a i , b i for i = 1 , . . . , g plus a loop c s around each s ∈ S , subject to the single relation � [ a 1 , b 1 ] · · · [ a g , b g ] c s = 1 . s ∈ S 1 The correct upper bound is 6(# S + 2 g − 2), but this requires some extra effort. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 3 / 37

Motivation: review of Bogomolov-Zhang Lifting in fundamental groups Since SL 2 ( R ) acts on R / 2 π Z via rotation, we obtain central extensions 1 → Z → � SL 2 ( Z ) → SL 2 ( Z ) → 1 , 1 → Z → � SL 2 ( R ) → SL 2 ( R ) → 1 . The former defines an element of H 2 (SL 2 ( Z ) , Z ), which we restrict along ρ : lift ρ ( a i ) , ρ ( b i ) , ρ ( c s ) to α i , β i , γ s ∈ � SL 2 ( Z ), so that � [ α 1 , β 1 ] · · · [ α g , β g ] γ s = m for some m ∈ Z . s ∈ S Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 4 / 37

Motivation: review of Bogomolov-Zhang Measurement in fundamental groups: part 1 For α ∈ � SL 2 ( R ), define its length as ℓ ( α ) = sup | α ( x ) − x | . x ∈ R By taking α i , β i , γ s to be “minimal” lifts of ρ ( a i ) , ρ ( b i ) , ρ ( c s ), we may ensure that ℓ ([ α i , β i ]) ≤ 2 π, ℓ ( γ s ) < π. Taking lengths of both sides of the equality � [ α 1 , β 1 ] · · · [ α g , β g ] γ s = m s ∈ S yields 2 π g + π # S > 2 π | m | = ⇒ 2 | m | ≤ 2 g + # S − 1 . Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 5 / 37

Motivation: review of Bogomolov-Zhang Measurement in fundamental groups: part 2 The group SL 2 ( Z ) is generated by these two elements: � 1 � � 1 � 1 0 u = , v = . 0 1 − 1 1 v ∈ � These may be lifted to elements ˜ u , ˜ SL 2 ( Z ) which generate freely modulo the braid relation ˜ u ˜ v ˜ u = ˜ v ˜ u ˜ v . We thus have a homomorphism deg : � SL 2 ( Z ) → Z taking ˜ u , ˜ v to 1. Write the discriminant divisor as ∆ E = � s ∈ S ∆ s · ( s ). then ρ ( c s ) ∼ u ∆ s . u ∆ s . Since Z ⊂ � Because of the minimal lifting, we have γ s ∼ ˜ SL 2 ( Z ) is v ) 6 , we have generated by (˜ u ˜ deg ∆ E = deg( m ) = 12 | m | . Comparing with the previous slide yields the claimed inequality. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 6 / 37

Motivation: review of Bogomolov-Zhang The role of theta This argument can be viewed as a prototype for IUT, but it is better to modify it first. We are implicitly using the interpretation E p ∼ = C / ( Z + τ p Z ). By exponentiating, we get E p ∼ p for q p = e 2 π i τ p . = C × / q Z This isomorphism can be described in terms of the a Jacobi theta function � ( − 1) n q ( n +1 / 2) 2 z 2 n +1 ϑ p ( z ) := p n ∈ Z via the formula ℘ p ( z ) = ( − log ϑ p ( z )) ′′ + c . In particular, this provides access to the symplectic structure in the guise of the Weil pairing. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 7 / 37

Motivation: review of Bogomolov-Zhang The role of nonarchimedean theta The previous discussion admits a nonarchimedean analogue for curves with split multiplicative reduction via Tate uniformization (e.g., see Mumford’s appendix to Faltings-Chai). The nonarchimedean theta function can be used to construct “something” (the tempered Frobenioid ) playing the role of γ s , i.e., which records the contribution of a bad-reduction prime to conductor and discriminant. We cannot hope to do this using only the profinite (local arithmetic) ´ etale fundamental group: this only produces the discriminant exponent as an element of � Z , whereas we need it in Z in order to make archimedean estimates. (Compare the analogous issue in global class field theory.) Fortunately, for nonarchimedean analytic spaces, there is a natural way to “partially decomplete” the profinite ´ etale fundamental group to obtain the tempered fundamental group (see Lepage’s lecture). Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 8 / 37

Construction of the ´ etale theta classes [EtTh, § 1] Context: Uniformization of elliptic curves Let K be a finite extension of Q p with integral subring o K . Let E be the analytification of an elliptic curve over K with split multiplicative reduction; it admits a Tate uniformization E ∼ = G m / q Z for some q ∈ K with v K ( q ) = v K (∆ E / K ) > 0 . We will describe a sequence of objects over E which admit analogues on the side of formal schemes (by comparison of fundamental groups); we denote this imitation by passing from ITALIC (or sometimes MAT HCAL ) to FRAKTUR lettering. The reverse passage is the Raynaud generic fiber construction. For example, for X as on the next slide, let X be the stable model of X over o K with log structure along the special fiber. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 10 / 37

Construction of the ´ etale theta classes [EtTh, § 1] Fundamental groups: setup denote 2 the profinite, topological, and tempered π, π top 1 , π tm Let � 1 fundamental groups of an analytic space (for some basepoint). Recall the exact sequence (on which G K acts): � π top � � � π tm � 1 1 π 1 ( G m , K ) 1 ( E K ) 1 ( E K ) � � � π tm � 1 . � Z 1 Z (1) 1 ( E K ) Let X be the hyperbolic log-curve obtained from E by adding logarithmic structure at the origin (the “cusp”). We have another exact sequence � π tm � π tm � G K � 1 . 1 1 ( X K ) 1 ( X ) 1 ( X K ) ell := π tm 1 ( X K ) ab := π tm Put π tm 1 ( E K ) and similarly for � π 1 . 2 In [EtTh], the letters Π and ∆ are generally used to distinguish arithmetic vs. geometric fundamental groups; hence the notation ∆ Θ on the next slide. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 11 / 37

� � � Construction of the ´ etale theta classes [EtTh, § 1] Fundamental groups: geometric Θ The group � π 1 ( X K ) is profinite free on 2 generators. Consequently, if we π 1 ( X K ) Θ for the 2-step nilpotent quotient write � π 1 ( X K ) Θ := � � π 1 ( X ) / [ • , [ • , • ]] , we have natural exact sequences (the second pulled back from the first): � ∆ Θ ∼ = � � π tm � 1 1 ( X K ) Θ π tm 1 ( X K ) ell 1 Z (1) � ∧ 2 � � � � � � 1 π 1 ( X K ) ab π 1 ( X K ) Θ π 1 ( X K ) ell 1 In fact, we can write     � � � � 1 Z (1) Z (1) 1 Z (1) Z (1) π 1 ( X K ) Θ ∼ 1 ( X K ) Θ ∼   , π tm   . � � = = 1 1 Z Z 1 1 Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 12 / 37

� � � Construction of the ´ etale theta classes [EtTh, § 1] Fundamental groups: covers and arithmetic Θ For any (possibly infinite) connected tempered cover W of X , π tm 1 ( W ) is an open subgroup of π tm 1 ( X ). We obtain exact sequences � ∆ Θ ∩ π tm � π tm � 1 1 ( W K ) Θ π tm 1 ( W K ) ell 1 1 ( W K ) � ∆ Θ ∩ π tm � � � � � 1 π 1 ( W K ) Θ π 1 ( W K ) ell 1 1 ( W K ) by taking subobjects of the corresponding objects over X K . Let L ⊆ K be the integral closure of K in the function field of W . Define 1 ( W ) Θ as the quotient of π tm π tm 1 ( W ) for which 1 ( W K ) Θ → π tm 1 ( W ) Θ → G L → 1 1 → π tm π 1 and/or • Θ replaced by • ell . is exact. Similarly, with π tm replaced by � 1 Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 13 / 37

Construction of the ´ etale theta classes [EtTh, § 1] The universal topological cover Let Y be a copy of G m with log structure at q Z , viewed as an infinite ´ etale cover of X . The exact sequence 1 ( Y K ) Θ → π tm 1 ( Y K ) ell → 1 1 → ∆ Θ → π tm consists of abelian 3 profinite groups; that is,   � � 1 Z (1) Z (1) π 1 ( Y K ) Θ = π tm 1 ( Y K ) Θ ∼   . � = 1 0 1 3 This is why we replaced • ab with • ell earlier. Kiran S. Kedlaya (UC San Diego) The ´ etale theta function 14 / 37