[IUTch-III-IV] from the Point of View of Mono-anabelian Transport III

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1 . [IUTch-III-IV] from the Point of View of Mono-anabelian Transport III. Theta Values Yuichiro Hoshi RIMS, Kyoto University July 22, 2016 Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

2 3.1 Kummer-compatible Multiradial Theta Functions 3.2 Local Logarithmic Gaussian Procession Monoids 3.3 log-kummer Correspondence III Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

3 3.1 Kummer-compatible Multiradial Theta Functions In order to establish a Kummer-compatible multiradial representation [whose coric data is F µ ] of theta values {q j2 v } l j=1 I : ( 3.1) we establish a multiradial representation of theta functions, ( 3.2) obtain {q j2 v } l j=1 I by applying the Galois evaluation operations to the multiradial theta functions, and ( 3.3) consider a log-kummer correspondence concerning {q j2 v } l j=1 I. Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

4 .. HT Θ±ellNF : a Θ ±ell NF-Hodge theater By identifying Ψ cns ( D ) 0 with Ψ cns ( D ) F l and pulling back O O /O via O /O suitable N, one obtains an F -prime-strip F ( D ), hence also F -, F µ -prime-strips F ( D ), F µ ( D ), i.e, the étale-like holomorphic symmetrized constant portions. a Kummer poly-isomorphism F µ F µ ( D ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

5 By applying a mono-anabelian reconstruction algorithm to D, one obtains an F -prime-strip F ( D ), hence also F -, F µ -prime-strips F ( D ), F µ ( D ), i.e, the étale-like mono-analytic symmetrized constant portions. a natural poly-isomorphism F µ ( D ) F µ ( D ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

6 Recall: F env: the F -prime-strip determined by Ψ Fenv ( HT Θ ) [i.e., at v V bad, F env,v: the Frob.-like theta monoid O K v, ΘN v ] F env( D > ): the F -prime-strip determined by Ψ env ( D > ) [i.e., at v V bad, F env( D > ) v : the mono-theta env. ver. of the theta monoid O K v, ΘN v ] natural poly-isom. F a Kummer poly-isom. F env F env and F ( D ) F env( D > ) F env( D > ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

7 Recall: F env = ( C env, Prime( C env) V, F env, { ρ env,v } v V ): the F -prime-strip determined by the Frob.-like theta monoids F env( D > ) = (D env( D >), Prime(D env( D >)) V, F env( D > ), { ρ D env,v} v V ): the F -pr.st. det d by the mono-theta env. ver. of the theta monoids a Kummer poly-isomorphism F env F env( D > ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

8 Recall: For v V bad : the Frobenius-like splitting theta monoid Ψ F env ( HT Θ ) v Ψ Fenv ( HT Θ ) v the étale-like splitting theta monoid Ψ env( D > ) v Ψ env ( D > ) v [by the constant multiple rigidity] [i.e., µ(k v ) Θ Q 0 v O K v, ΘQ 0 v ] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

9 Ψ F env ( HT Θ ) v Ψ env( D > ) v Ψ F env ( HT Θ ) µ v Ψ Fenv ( HT Θ ) µ v Ψ env( D > ) µ v Ψ env ( D > ) µ v Ψ Fenv ( HT Θ ) v Ψ env ( D > ) v Ψ Fenv ( HT Θ ) µ v Ψ env ( D > ) µ v F µ,v F µ ( D ) v The composite from ( ) µ to F µ,v is zero. Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

10 The id on Π µ (M Θ ( D >,v )) Q/Z Ψ env ( D > ) µ v i.e., Ψ env( zero eval. D > ) v Ψ env ( D > ) µ v the splitting theta monoid, the cyclotomes related to theta functions, and the spl g Ψ env ( D > ) v /µ ( Ψ env( D > ) v /µ) Ψ env ( D > ) µ v, is compatible, relative to the above diagram, w/ Aut(F µ,v ). Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

11 In particular: R tht : the collection of data consisting of: (a) HT D-Θ±ell NF (b) F µ ( D ) (c) F env( D > ) (d) the full poly-isomorphism F µ env ( D > ) full F µ ( D ) for v V bad : (e) (f) the proj. sys. of mono-theta env. M Θ ( D >,v ) cons d from D >,v Π µ (M Θ ( D >,v )) Q/Z Ψ env ( D > ) µ v Ψ env( zero eval. D > ) v Ψ env ( D > ) µ v A morphism between R tht = an isom. of (a) [ isom. of (c), (e), (f)] and an isom. of (b) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

12 R tht F µ ( D ) is multiradial. In particular, HT Θ±ell NF Θ µ gau HT Θ±ell NF R tht Moreover, in the resulting isomorphism R tht R tht : F µ env ( D > ) F µ env ( D > ) is comp. w/ full full F µ ( D ) F µ ( D ) F µ env full F µ R tht F µ env full F µ ( f) ( f) is comp. w/ an isom. of the corresp g Frob.-like objects [relative to the Kmm poly-isom. and poly-isom. of mono-theta env.] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

13 3.2 Local Logarithmic Gaussian Procession Monoids.. HT Θ±ell NF log HT Θ±ellNF : a log-link v V non Ψ Fgau ( HT Θ ) v Ψ Fgau ( HT Θ ) v j J (Ψ F v ) j log j J Ψ log( F v ) j j J log( F v ) j j J (Ψ F v ) j [For v V bad : O K v, (q12 v,..., q (l ) 2 v ) N O K v, (q12 v,..., q (l ) 2 v ) Q 0 j J O K v,j j J O log(o ) j J log(o ) log K v,j j J O ] K v,j Kv,j Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

14 Ψ FLGP ( HT Θ±ell NF ) v = Im(Ψ Fgau ( HT Θ ) v j J log( F v ) j ) Ψ FLGP ( HT Θ±ell NF ) v = Im( Ψ Fgau ( HT Θ ) v j J log( F v ) j ) One may construct similar objects for v V arc. Ψ FLGP ( HT Θ±ell NF ) = {Ψ FLGP ( HT Θ±ell NF ) v } v V, Ψ FLGP ( HT Θ±ell NF ) = { Ψ FLGP ( HT Θ±ell NF ) v } v V : local logarithmic Gaussian procession monoids Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

15 F LGP : the F -prime-strip determined by Ψ FLGP ( HT Θ±ell NF ) [i.e., at v V bad, F LGP,v : the Frobenius-like Gaussian monoid O K v, (q12 v,..., q (l ) 2 v ) N in j J log(o K v,j ) ] F lgp = F LGP One obtains F µ -prime-strips F µ LGP F µ lgp F µ gau = ( C LGP, Prime( C LGP ) V, F µ LGP, { ρ LGP,v } v V ) = ( C lgp, Prime( C lgp ) V, F µ lgp, { ρ lgp,v } v V ) F µ LGP F µ lgp Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

16 . HT Θ±ellNF : a Θ ±ell NF-Hodge theater HT Θ±ell NF log. HT Θ±ell NF : a log-link F µ -prime-strips F µ, F µ, and F µ LGP lgp the Θ µ LGP -link HT Θ±ell NF Θ µ LGP HT Θ±ell NF the full poly-isomorphism F µ LGP F µ full the Θ µ lgp -link HT Θ±ell NF Θ µ lgp HT Θ±ell NF the full poly-isomorphism F µ lgp F µ full Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

17 a q-pilot object [of C ] an obj. of C det d by generators of the spl. of F,v [v Vbad ] [i.e., an object of C determined by (q v) v V bad] a Θ-pilot object [of C LGP or C lgp ] obj. of C LGP or C lgp det. by gen. of the spl. of F LGP,v [v Vbad ] [i.e., an object of C LGP or C lgp determined by (q12 v,..., q (l ) 2 v ) v V bad] For {LGP, lgp}, the poly-isom. C C induced by HT Θ±ell NF Θ µ HT Θ±ellNF maps a Θ-pilot object to a q-pilot object. Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

18 For {LGP, lgp}: a -Gaussian log-theta-lattice Θ µ log. n,m+1 HT Θ±ell NF log Θ µ n,m HT Θ±ell NF log Θ µ log. n+1,m+1 HT Θ±ell NF log Θ µ n+1,m HT Θ±ell NF log Θ µ Θ µ.. Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

19 full n,m HT D-Θ±ell NF full n,m+1 HT D-Θ±ell NF full [vertical] full n,m D full n,m+1 D full [vertical] full n,m D n,m+1 D [vertical] full full full n,m F µ full n,m D n+1,m F µ full full [horizontal] n+1,m D [horizontal] full full étale-like structure [i.e., HT D-Θ±ell NF ]: vertically coric Frobenius-like mono-an. structure [i.e., F µ ]: horizontally coric étale-like mono-analytic structure [i.e., D ]: bi-coric Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

20 3.3 log-kummer Correspondence III. log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log. HT D-Θ±ellNF : the ass d D-Θ ±ell NF-Hodge theater [up to isom.] v V non Ψ gau ( D ) v Ψ gau ( D ) v j J (Ψ cns( D ) j ) v [For v V bad : j J (log(f( D)) j ) v j J (Ψ cns( D ) j ) v O K v, (q12 v,..., q (l ) 2 v ) N O K v, (q12 v,..., q (l ) 2 v ) Q 0 j J O K v,j = j J O log(o Kv,j ) j J log(o K v,j ) log j J O K v,j ] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

21 Ψ LGP ( HT D-Θ±ell NF ) v = Im(Ψ gau ( D ) v j J (log(f( D )) j ) v ) Ψ LGP ( HT D-Θ±ell NF ) v = Im( Ψ gau ( D ) v j (log(f( D )) j ) v ) One may construct similar objects for v V arc. Ψ LGP ( HT D-Θ±ell NF ) = {Ψ LGP ( HT D-Θ±ell NF ) v } v V, Ψ LGP ( HT D-Θ±ell NF ) = { Ψ LGP ( HT D-Θ±ell NF ) v } v V : F ( HT D-Θ±ell NF ) LGP : the F -pr.-st. det d by Ψ LGP ( HT D-Θ±ell NF ) [i.e., at v V bad, F ( HT D-Θ±ell NF ) LGP,v : the étale-like Gaussian monoid O K v, (q12 v,..., q (l ) 2 v ) N in j J log(o K v,j ) ] F ( HT D-Θ±ell NF ) lgp = F ( HT D-Θ±ell NF ) LGP Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

22 One obtains F -prime-strips F ( HT D-Θ±ell NF ) LGP = ( C D LGP, Prime V, F ( HT D-Θ±ell NF ) LGP, { ρ D LGP,v } v V) F ( HT D-Θ±ell NF ) lgp = ( C D lgp, Prime V, F ( HT D-Θ±ell NF ) lgp, { ρ D lgp,v } v V) F ( HT D-Θ±ell NF ) gau F ( HT D-Θ±ell NF ) LGP F ( HT D-Θ±ell NF ) lgp Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

23 v V bad m Z, 1 j l Ψ F LGP ( m HT Θ±ell NF ) v,j I Q ( S± j+1,j;m F v ) 1 j l Ψ LGP ( HT D-Θ±ell NF ) v,j I Q ( S± j+1,j; D v ) Kummer [poly-]isomorphisms Ψ F LGP ( m HT Θ±ell NF ) v,j Ψ LGP ( HT D-Θ±ell NF ) v,j I Q ( S± j+1,j;m F v ) I Q ( S± j+1,j; D v ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

24 These [m Z] satisfy the following non-interference property : Let us consider the diagram Ψ F LGP ( m HT ) v,j I Q ( S± j+1,j;m F v ) Kmm log I Q ( S± j+1,j;m+1 F v ) Kmm log I Q ( S± j+1,j;m+2 F v ) Kmm log I Q ( S± j+1,j; D v ) I Q ( S± j+1,j; D v ) I Q ( S± j+1,j; D v ) The log [at m] is ined on O log( j;m F v). Ψ F LGP ( m HT Θ±ell NF ) v,j O log( j;m F v ) µ Ker(log). Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25

25 Let us think that Ψ F LGP ( m HT Θ±ell NF ) v,j acts on I Q ( S± j+1,j; D v ) [not via a single Kummer isomorphism which fails to be compatible with the sequence of log-links but rather] via the totality of Kmm log Z 0. Note: These are mutually compatible up to id at an adjacent m. One obtains a sort of log-kummer correspondence between the totality of { Ψ F LGP ( m HT Θ±ell NF ) v,j }m Z and their actions on I Q ( S± j+1,j; D v ). ( m C LGP, {m ρ LGP,v } v V ) ( C D LGP, { ρ D LGP,v } v V) [m Z] are mutually compatible. Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 25