. Hodge-Arakelov-theoretic Evaluation I. Yuichiro Hoshi. December 10, 2015

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1 .. Hodge-Arakelov-theoretic Evaluation I Yuichiro Hoshi RIMS, Kyoto University December 10, 2015 Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

2 Notation and Terminology : the geometric portion of Π, i.e., the kernel of the outer surjection from Π to the arithmetic quotient of Π ( ): the profinite completion of ( ) For a topological group G, H i (G, A) def = lim H G: open subgps of finite index H i (H, A) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

3 2 Galois-theoretic Theta Evaluation In 2, 2 1 2, and 3: Fix a v Vbad. def Π v = Π tp func l X v alg m Π tp Ÿ v Π tp Ÿ v Π tp Y v Π v = Π tp X v Π tp Y v Π ± v Π cor v def = Π tp X v def = Π tp C v M Θ = (M Θ M ) M: a projective system of mono-theta environments F v : the tempered Frobenioid determined by X v Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

4 Goal: Θ v evaluation labeled by t q t2 v (t LabCusp ± at v) Problem 1 A conjugacy indeterminacy in a situation related to the theta value q t2 at t F l depends, a priori, on the label t F l. v That is to say, various objects at t F l is well-defined up to conjugation which is, a priori, independent of the label t F l. On the other hand, we want to establish a suitable Kummer theory for such theta values. We have to synchronizes conjugacy indeterminacies in situations related to the theta values at various t F l. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

5 Problem 2 By using the structure of a Hodge-theater, we synchronized globally the various LabCusp ± v by means of relative to through 0 D,v D ± ( through 0 D,w ) the var. bij. {cuspidal inertia subgps of π tp 1 (X v )}/Inn(π tp 1 (X v )) {cuspidal inertia subgps of πét 1 (X K )}/Inn(πét 1 (X K )). Moreover, the crucial global F ± l -symmetry arises from a profinite conjugation (cf. πét 1 (C F )/πét 1 (X F ) πét 1 (C K )/πét 1 (X K ) = F ± l ). We have to discuss the comparison between tempered conjugation and profinite conjugation. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

6 Γ ( ) : the dual graph of the special fiber of ( ) v Irr( ): the set of irreducible components of the special fiber of ( ) v Fix an inversion automorphism ι def = ιÿ of Ÿ v ι X, ιÿ, and Z Irr(Ÿ ) ( = Irr(Y ) = Irr(Ÿ ) = Irr(Y )), (Note: ι Irr 1 Z) a subgraph Γ ( ) Γ ( ) for ( ) {X, X, Ÿ, Ÿ }, i.e., l l l 1 l t LabCusp ± (Π v ) a subgraph Γ t def t l ( ) Γ ( ) l +1 for ( ) {X, X, Ÿ, Ÿ }, i.e., l 1 l = 0 (i.e., for instance, Γ ( ) def = Γ 0 ( ) ) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

7 { t, } Π v Π v : a decomposition subgroup of Γ X Γ X, i.e., Π tp X,Γ X Π ± v Π v def = N Π ± v (Π v ) Π ± v def = Π v Π tp Ÿ v Π tp Ÿ v Thus, for instance: Π ± v /Π v Π ± v /Π v Gal(X v /X v ) ( = µ l ), Π ± v Π v = Π v [Π v : Π v ] = 2, [Π± v : Π v ] = l,... Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

8 Then: func l Π v alg m (Π v Π v Π v, ι) well-defined up to Π v -conj. (Recall: Π v func l alg m θ(π v) θ(π v ) H 1 (Π tp Ÿ (Π v), (l Θ )(Π v ))) Thus: func l Π v alg m θι (Π v ) θ(π v ), θ ι (Π v ) θ(π v ): µ 2l -, µ-torsors func l Moreover: Π v alg m (l Θ )(Π v ): the subquotient of Π v det d by (l Θ )(Π v ) Π v G v (Π v ): the arithmetic quotient of Π v Π v G v (Π v ): the arithmetic quotient of Π v Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

9 . Key Lemma (Comparison Between Temp d Conj. and Prof. Conj.). I t Π v : an inertia subgp ass d to the cusp lab d by t s.t. I t v γ, γ ± v Then the following three conditions are equivalent: γ ± v I (γ γ ) t Π γ v I (γ γ ) t (Π ± v )γ where. ( ) γ def = γ ( ) γ 1 Key Lemma follows from the theory of semi-graphs of anabelioids. In the situation of Key Lemma, write δ def = γ γ ± v. ( Lemma It δ = I (γ γ ) t Π γ v = Πδ v ) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

10 By the theory of semi-graphs of anabelioids, one can construct, from the inclusions I δ t = I (γ γ ) t (a) the dec. gp D δ t def = N Π δ v (I δ t ) Π δ v that contains I δ t Π γ v = Πδ v : (b) a dec. gp D δ µ Π δ v, well-def d up to (Π± v )δ -conj., ass d to µ (c) a dec. gp D δ t,µ Π δ v, well-def d up to (Π± v )δ -conj., ass d to the µ -translation of the cusp that det. I δ t (i.e., an ev. pt lab d by t) Moreover, this construction is compatible w/: the ± v -conjugation the inclusion Π v t Π v If, moreover, = t, then the construction of (a) and (c) is compatible w/ the cor Π v -conjugation. (Recall: the F ± l -symmetry arises from the cor Π v -conjugation.) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

11 . I. t δ = I (γ γ ) t Π δ v Πγ v = Π δ v By restricting θ ι (Π γ v) θ ι (Π γ v) to Π γ v Πtp Ÿ (Πγ v), we obtain µ 2l -, µ-torsors θ ι (Π γ v ) θ ι (Π γ v ) H 1 (Π γ v, (l Θ)(Π γ v )). Thus, by restricting them to (G v (Π γ v ) ) Dt,µ δ Π γ, we obtain v θ t (Π γ v ) θ t (Π γ v ) H 1 (G v (Π γ v ), (l Θ)(Π γ v )), i.e., µ 2l q t2 v µ q t2 v, where l t l is det d by t. ( )θ t (Π γ v ) def = ( ) θ t (Π γ v ) (= ( )θ t (Π γ v )) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

12 In summary: func l Π v alg m {θ t (Π γ v )} t F l, { θ t (Π γ v )} t F l arising from Π γ v : an arbitrary ± v -conjugate of Π v It δ : an arbitrary ± v -conjugate of I t s.t. It δ Π γ v γ-conj. (t ranges over the elements of LabCusp ± (Π γ v) LabCusp ± (Π v )) Moreover, this alg m is compatible w/ the independent conj. actions of ± v on the sets of (not temp d but) prof. conj. {Π γ v } γ and {I δ t } δ. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

13 . Remark. A conjugacy indeterminacy in a situation related to the theta value ( ) θ t at t F l depends, a priori, on the label t F l. That is to say, various objects at t F l is well-defined up to conjugation which is, a priori, independent of the label t F l. However, our resulting theta values ( )θ t (Π γ v ) H 1 (G v (Π γ v ), (l Θ)(Π γ v )) for various t F l are computed relative to label-independent G v (Π γ v ) and (l Θ)(Π γ v ). conjugate synchronization.( One may apply Kummer theory related to theta values.) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

14 func l Recall: Π v M TM (Π v) (M TM alg m θι )(Π v ) (M TM θ ι )(Π v ) in H 1 (Π v, (l Θ )(Π v )) (Recall: M TM is an isomorph of O F v.) By restricting them to Π γ v Π v, we obtain M TM (Πγ v ) (M TM θι )(Π γ v ) (M TM θ ι )(Π γ v ) in H 1 (Π γ v, (l Θ)(Π γ v )) Thus, by the nat l θ ι (Π γ v ) θ 0 (Π γ ), we obtain a splitting v (M TM θ ι )(Π γ v )/M µ TM (Πγ v ) = M µ TM (Πγ v ) ( θ ι (Π γ v )/M µ TM (Πγ v ) ). (Recall: M µ µ TM (resp. M TM ) is an isomorph of Oµ (resp. O µ ).) F v F v Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

15 In the remainder of 2, suppose: Π tp X (MΘ ) = Π v 1 Π µ (M Θ ) Π M Θ Π tp X (MΘ ) By base-chan g Π v Π v Π v = Π tp X (MΘ ) via Π M Θ Π tp X (MΘ ), we obtain closed subgroups Π M Θ Π M Θ Π M Θ. Π µ (M Θ ), (l Θ)(M Θ ), Π v (M Θ ), G v(m Θ ): the respective corresponding subquotients of Π M Θ a cyclotomic rigidity isomorphism (l Θ )(M Θ ) Π µ (M Θ ) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

16 By applying the cycl. rig. isom. (l Θ )((M Θ )γ ) Π µ ((M Θ )γ ) arising from the γ-conjugate (M Θ ) γ of M Θ θ ι (Π γ v ) θ ι (Π γ v ) H 1 (Π γ v, (l Θ)(Π γ v )), θ t (Π γ v ) θ t (Π γ v ) H 1 (G v (Π γ v ), (l Θ)(Π γ )), and v (where γ ± v ) to (M TM θ ι )(Π γ v )/M µ TM (Πγ v ) = M µ TM (Πγ v ) ( θ ι (Π γ v )/M µ TM (Πγ v )), we obtain θ ι env ((MΘ )γ ) θ ι env ((MΘ )γ ) H 1 (Π v ((M Θ )γ ), Π µ ((M Θ )γ ), θ t env ((MΘ )γ ) θ t env ((MΘ )γ ) H 1 (G v ((M Θ )γ ), Π µ ((M Θ )γ )), (M TM θ ι env )((MΘ )γ )/M µ TM ((MΘ )γ ) = M µ TM ((MΘ )γ ) ( θ ι env ((MΘ )γ )/M µ TM ((MΘ )γ ) ). Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

17 In a similar vein, by applying the cycl. rig. isom. (l Θ )((M Θ )γ ) µẑ(g v ((M Θ )γ )), we obtain: θ ι bs ((MΘ )γ ) θ ι bs ((MΘ )γ ) H 1 (Π v ((M Θ )γ ), µẑ(g v ((M Θ )γ ))), θ t bs ((MΘ )γ ) θ t bs ((MΘ )γ ) H 1 (G v ((M Θ )γ ), µẑ(g v ((M Θ )γ ))), (M TM θ ι bs )((MΘ )γ )/M µ TM ((MΘ )γ ) bs ) = M µ TM ((MΘ )γ ) bs ( θ ιbs ((MΘ )γ )/M µtm ((MΘ )γ ) bs. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

18 2 1 2 Multiradial Kummer-detachment of Theta Monoids Goal: multiradial Kmm-detach. of theta monoids, i.e., O F v Θ v Strategy: By the final assertion of 1, we have: Π v multiradial alg m via multiradial cycl. rig. étale-like O F v Θ v mono-theta O F v Θ v, i.e., labeled by env Thus, by applying the Kummer theory for theta functions: Frobenius-like O Kummer Θ F v v theory (O F v Θ v ) env Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

19 func l Recall: F v µ 2l(T ) Θ alg m Ÿ v v O (T ) Ÿ v (which determines the monoid O Cv Θ { Ψ F Θ v = Ψ F Θ v = Ψ F Θ v,α func l Recall: F v alg m } def = O (A Θ C ) (Θ α v Θ v )N A Θ α Aut D v (Ÿ ) v (A Θ ) = O (A Θ C ) Θ N v Θ v A Θ ) { } def Ψ F Θ v,α = O (A Θ C ) (Θ α v Θ v )Q 0 A Θ α Aut D v (Ÿ ) v the base-theoretic hull C v F v def (Π v ) Ψ Cv = OC v (A Θ ) (well-defined up to Π v -conjugation). ( )Ψ F Θ v : the Frobenius-like theta monoid. Ψ Cv : the Frobenius-like constant monoid Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

20 Recall: M Θ func l M TM alg m (MΘ ), θ env (M Θ ) θ env (M Θ ) in H 1 (Π tp Ÿ (MΘ ), Π µ (M Θ )) { } Ψ env (M Θ ) def = Ψ ι env(m Θ ) def = M TM (MΘ ) θ ι env (MΘ ) N { } Ψ env (M Θ ) def = Ψ ι env(m Θ ) def = M TM (MΘ ) θ ι env (MΘ ) Q 0 ι: inv. autom. Recall: M Θ func l G v(m Θ ), µẑ(g v (M Θ )) Π µ (M Θ ) alg m func l M TM(M Θ ) H 1 (Π tp alg m Ÿ (MΘ ), Π µ (M Θ )) (Recall: M TM is an isomorph of O.) F v (Π tp X (MΘ ) ) Ψ cns (M Θ ) def = M TM (M Θ ). ( )Ψ env (M Θ ): the mono-theta-theoretic theta monoid. Ψ cns (M Θ ): the mono-theta-theoretic constant monoid ι: inv. aut. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

21 In particular, by applying the above algorithm to M Θ (Π v ):. ( )Ψ env (M Θ (Π v )): the étale-like theta monoid. Ψ cns (M Θ (Π v )): the étale-like constant monoid In order to obtain multiradial Kummer-detachment of Θ v, let us relate Frobenius-like/mono-theta-theoretic/étale-like theta monoids. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

22 In the remainder of 2 1 2, suppose: MΘ (F v ) = M Θ Then, by applying the Kummer theory, relative to a suitable assign t ι α, we obtain an isomorphism ( ) Ψ F Θ v,α ( ) Ψ ι env(m Θ ) (cf. the Kummer theory of theta functions in tempered Frobenioids). Write ( )Ψ F Θ v ( ) Ψ env (M Θ ) for the collection of the above isomorphisms. Moreover, again by applying the Kummer theory, we obtain an isomorphism Ψ Cv Ψ cns (M Θ ). Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

23 Thus, every isomorphism M Θ (Π v ) M Θ = M Θ (F v ) determines: (étale) (mono-theta) (Frobenius) Π v = Π tp X (MΘ (Π v )) Π tp X (MΘ ) = Π tp X (MΘ ) Ψ env (M Θ (Π v )) Ψ env (M Θ ) Ψ F Θ v Ψ env (M Θ (Π v )) Ψ env (M Θ ) Ψ F Θ v G v = G v (M Θ (Π v )) G v (M Θ ) = G v (M Θ ) Ψ env (M Θ (Π v )) Ψ env (M Θ ) Ψ F Θ v Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

24 Moreover, every isom. M Θ (Π v ) M Θ = M Θ (F v ) also determines: (étale) (mono-theta) (Frobenius) Π v = Π tp X (MΘ (Π v )) Π tp X (MΘ ) = Π tp X (MΘ ) Ψ cns (M Θ (Π v )) Ψ cns (M Θ ) Ψ Cv G v = G v (M Θ (Π v )) G v (M Θ ) = G v (M Θ ) Ψ cns (M Θ (Π v )) Ψ cns (M Θ ) Ψ C v That is to say, we obtain various Kummer isomorphisms. Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

25 Thus, by the final assertion of 1, we obtain multiradial Kummer-detachment of theta monoids: Π v multiradial alg m (cf. 1) ( ) Ψ env (M Θ (Π v )) via multiradial cycl. rig.. Remark. On the other hand, the above discussion only gives uniradial Kummer-detachment of constant monoids.. ( )Ψ F Θ v (cf. (Π v Ψ Cv ) (G v (M Θ ) Ψ C v ) : a uniradial environment) ( the theory of log-shells) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

26 2 3 4 Definition (used in 4) (1) v V non Fv : the Fro d corresponding to G v O F v (omit the case of v V arc ) an F -prime-strip def {an isomorph of F v } v V (2) v V non Fv µ : the µ-kummer Frobenioid corresponding to G v O µ F v {Im ( (O F v ) H = O F H v equipped with the µ-kummer structure, i.e., (omit the case of v V arc ) O F v O µ F v ) }H Gv: open subgps an F µ -prime-strip def {an isomorph of F µ v } v V Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

27 (3) F = { F v } v V : an F -prime-strip v V bad Fv corresponds to G v (O q N mod µ 2l F q N v v : the split- µ-kummer Frobenioid corresponding to F µ v G v (O µ F v (µ 2l q N v /µ 2l) (µ 2l q N v /µ 2l)) w/ µ-kmm str. v V good V non F v corresponds to G v (O F v p N v p N v ) Fv µ : the split- µ-kummer Frobenioid corresponding to G v (O µ F v p N v p N v ) w/ µ-kmm str. (omit the case of v V arc ) an F µ -prime-strip def {an isomorph of F µ v } v V v ) Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28

28 (4) an F µ -prime strip def a suitable collection of data ( C, Prime( C ) V, F µ, { ρ v } v V ), i.e., a collection of data obtained by replacing the F of an F -prime strip ( C, Prime( C ) V, F, { ρ v } v V ) by an F µ -prime-strip F µ Thus: F -prime-strip func l alg m F -prime-strip func l alg m F -prime-strip func l alg m F -prime-strip func l alg m F µ -prime-strip F µ -prime-strip F µ -prime-strip Yuichiro Hoshi (RIMS, Kyoto University) Hodge-Arakelov-theoretic Evaluation I December 10, / 28