Go Yamashita. 11/Dec/2015 at Oxford

Transcription

1 IUT III Go Yamashita RIMS, Kyoto 11/Dec/2015 at Oxford The author expresses his sincere gratitude to RIMS secretariat for typesetting his hand-written manuscript. 1 / 56

2 In short, in IUT II, we performed Galois evaluation theta fct theta values env labels gau labels (MF -objects (filtered φ-modules) Galois rep ns) 2 / 56

3 Two Problems 1. Unlike theta fcts, theta values DO NOT admit a multiradial alg m in a NAIVE way. 2. We need ADDITIVE str. for (log-) height fcts. µ log 3 / 56

4 On 1. Gal. eval. theta fcts theta values admit multirad. alg m ( cf. [IUT II, Prop 3.4(i)] ) ( constant monoids F ) l requires cycl. rig. via LCFT (cf. [IUT II, Prop 3.4(ii), Cor 3.7(ii), Rem (ii)]) 4 / 56

5 Recall cycl. rig. via LCFT uses O = (unit portion) (value gp portion) O core O theta values We DO NOT share it in both sides of Θ-link! {q j2 } j q DO NOT admit a multirad. alg m in a NAIVE way. 5 / 56

6 cf. use only µ-portion O µ 0 O µ cycl. rig. via mono-theta env. cycl. rig. via Ẑ Q >0 = {1} µ do not interfere core µ 0 7 admit a multirad. alg m 6 / 56

7 To overcome these problems, use log link! & allowing mild indet s non-interference etc. (later) 7 / 56

8 want to see alien ring str. O µ Θ-link log-link eye Note F ± l -symm. isom s are compatible w/ log-links can pull-back Ψ gau via log-link 8 / 56

9 However, O µ Θ log log Θ is highly non-commutative (cf. log(a N ) (log a) N ) cannot see from the right log 9 / 56

10 We consider the infinite chain of log-links log log log log this is invariant by one shift! 10 / 56

11 Important Fact k/q p fin. log O k log O k 1 2p log O k log shell = I k the domain & codomain of log are contained in the log-shell upper semi-compatibility (Note also: log-shells are rigid) 11 / 56

12 Besides theta values, we need another thing : we need NF (:= number field) to convert -line bdles into -line bdles and vice versa. 12 / 56

13 -line bdles def d in terms of torsors -line bdles def d in terms of fractional ideals natural cat. equiv. in a scheme theory 13 / 56

14 -line bdles def d only in terms of -str s admits precise log-kummer corr. But, difficult to compute log-volumes -line bdles def d by both of & -str s only admits upper semi-compatible log-kummer corr. But, suited to explicit estimates 14 / 56

15 We also include NFs as data (an NF) j v Q log(o ) theta values NFs } story goes in a parallel way in some sense ( of course essential difference cf. [IUT III, Rem 2.3.2, 2.3.3] ) 15 / 56

16 To obtain the final multirad. alg m: Frob.-like data assoc. to F-prime-strips Kummer theory étale-like data assoc. to D-prime-strips arith.-hol. forget arith. hol. str. mono-an. data assoc. to D -prime-strips 16 / 56

17 wall étale transport étale-like Kummer detachment Θ Frob.-like 17 / 56

18 Frobenius-picture log log Kummer log log theory log Θ log (cycl. rig. s) core étale-picture rad. data permutable rad. data 18 / 56

19 3 portions of Θ-link local G v G v share ( ht + fct) unit O µ O µ value gp q 1 2 N j 2 q N drastically changed global realified V v (R 0 ) v (, j 2, ) (R 0 ) v log q ( ht fct) 19 / 56

20 Kummer theory unit portions G v O µ = O k /µ Q p -module Θ wall + integral str. i.e. Im(O k ) (O µ H k )H fin. gen. H G v Z p mod. open non-ring theoretic log-shell G v O µ ( computable log-vol. ) 20 / 56

21 ( G v O ) k Kummer ( G v O k ( G v )) unlike the case of O k, Ẑ - indet. occurs container is invariant under this Ẑ -indet. OK cycl. rig. µ(g v ) µ(o k ) via LCFT? does not hold. now, we cannot use O k. use only O k 21 / 56

22 We want to protect value gp portion global real d portion from this Ẑ - indet! sharing O µ O µ w/ int. str. (Ind 2) Θ horizontal indet. 22 / 56

23 value gp portion O mono-theta cycl. rig. (unlike LCFT cycl. rig.) only µ is involved shared O µ O µ 0 core 0 µ µ NF portion Ẑ Q >0 = {1} cycl. rig. do not obstruct each others multirad. (on the function level) 23 / 56

24 Note also mono-theta cycl. rig. is compat. w/ prof. top. F ± l - sym. (conj. synchro.) log F ± l - sym. is compat. w/ log-links can pull-back coric (diagonal) obj. via log-links later LGP monoid (Logarithmic Gaussian Procession) 24 / 56

25 log log value gp portion After Kummer (Kummer) (log) n (n 0), take the action of q j2 on I Q coric Kummer log-kummer correspondence unit portion not compat. Kummer consider of a common rigid upper bound given by log-shell (Ind 3) log vertical indet. 25 / 56

26 value gp portion const. multiple rig. log label 0 splitting modulo µ of hor. core 0 O O q j2 O q j2 /O 0 & logp (µ) = 0 No new action appears by the iteraions of log. s No interference 26 / 56

27 Note also µ log (log p (A)) = µ log (A) if A bij log p (A) compatibility of log-volumes ( w/ log-links ) do not need to care about how many times log. s are applied. 27 / 56

28 In the Archimedean case, we use a system (cf. [IUT III, Rem 4.8.2(v)]) & { O /µ N O /µ N } µ N is killed in O /µ N & constructions (of log-links, ) & start from O /µ N s, not O (cf. [IUT III, Def 1.1(iii)]) we put weight N on O /µ N for the log-volumes (cf. [IUT III, Rem.1.2.1(i)]) 28 / 56

29 NF portion as well, consider the actions of (F mod ) j after (Kummer) (log) n (n 0) By F mod v O v = µ No new action appears in the interation of log. s No interference 29 / 56

30 cf multirad. contained in geom. container a mono-analytic container val gp eval theta fct theta values depends on labels ( & hol. str. ) q j2 NF eval ( )κ-coric fcts NF indep. of labels ( dep. on hol. str. ) F mod (up to {±1}) Belyǐ cusp tion 30 / 56

31 cycl. rig log-kummer theta mono-theta cycl. rig. no interference by const. mult. rig. NF Ẑ Q >0 = {1} no interference cycl. rig by F mod v O v = µ 31 / 56

32 vicious cycles µ Fr log Kummer Kummer µ ét µ ét indet. = I ord N 1 {±1} µ Fr theta I ord = {1} by zero of order = 1 at each cups cf. q j 2 s are NF I ord Im N 1 not inv. log under {±1} {1} by Ẑ Q >0 = {1} However the totally of we have (F OK. F mod ) {±1}-indet. mod is invariant under {±1} 32 / 56

33 cf. [IUT III, Fig 2.7] 0 is also permuted F ± l - sym. theta fct local & transcendental zero of order = 1 at each cusp theta q = e 2πiz only one valuation compat. w/ prof. top. cycl. rig. NF F l -sym. 0 is isolated global & algebraic rat. fcts. Note theta fcts/ theta values do not have F ± l - sym. But, the cycl. rig. DOES. use [, ] / 56 Never for alg. rat. fcts incompat. w/ prof. top. Ẑ Q >0 = {1} sacrifice the compat. w/ prof. top. many valuations global

34 Note also Gal. eval. use hol. str. labels theta Gal. eval. & Kummer compat. w/ labels the output F mod does not depend on labels. NF global real d monoids are mono-analytic nature ( units are killed) do not depend on hol. str. 34 / 56

35 unit O µ O µ (Ind 2) val. gp {q j2 } w/ (Ind 3) I Q ( ) NF M mod Kummer detachement étale-like objects 35 / 56

36 étale transport full poly G v G v (Ind 1) permutative indet. we can transport the data over the Θ-wall 36 / 56

37 Another thing Ψ gau t F l (const. monoids) labels come from arith. hol. str / 56

38 cannot transport the labels for Θ-link Θ-link F l 1 2 l ( = l 1 2 )??? 38 / 56

39 0? l ± -possibilities 1? l? in total (l ± ) l± -possibilities (l ± = l + 1 = l+1 2 ) the final inequality is weak. 39 / 56

40 use processions {0} {0, 1} {0, 1, 2} {0,, l } {?} {?,?} {?,?,?} {?,,?} then, in total (l ± )!-possibilities gives more strict inequality than the former case 40 / 56

41 A rough picture of the final multirad. rep n: (F mod ) 1 (F mod ) l {I Q 0 } {IQ 0, I Q 1 } {IQ 0,, I l Q } q 12 q 22 acts q (l ) 2 Ψ LGP const. mult. rig. value gp portion via canonical splitting modulo µ 41 / 56

42 By this multirad. rep s & the compatibility w/ Θ-link : {q j2 } 1 j l q w/ indet. s we cannot distinguish them!! deg 0 deg 0 w/ indet. s 42 / 56

43 (Ind1) permutative indet. G v G v in the étale transport (Ind2) horizontal indet. Θ O µ O µ in the Kummer detach. w/ int. str. (Ind3) vertical indet. log in the Kummer detach. log(o ) log O 1 2p log(o ) can be considered as a kind of descent data from Z to F 1 43 / 56

44 Z F1 (Ind1) Z hol. hull (Ind2) (Ind3) 44 / 56

45 q mono-analytic container log-shell hol. hull I Q possible images of {q j2 } j somewhere, it contains a region with the same log-volume as q 45 / 56

46 Recall {q j2 } j q 0 (ht) + (indet) log -diff ( 1 + ε) ( (+ log -cond) ) (ht) (1 + ε)(log -diff + log -cond) calculation in Hodge-Arakelov miracle equality 1 l 2 j j 2 [ log q ] l2 24 [ log q ] )) 1 l j[ ω E ] l2 24 [ log q ] 46 / 56

47 cf. Hodge-Arakelov IF a global mult. subspace existed qo q j2 O deg 0 deg 0 (large) 0 47 / 56

48 [IUT III, Th 3.11] In summary, tempered conj. ( diagonal hor. core ) vs prof. conj. F ± l -conj. synchro (semi-graphs of anbd.) compat. w/ (i)(objects) (ii)(log-kummer) (iii) ( Θ µ LGP -link ) F ± -symm. l F l -symm. others I unit Ψ LGPval gp compat. of log-link w/ F ± l -symm. ( ) M mod NF Belyǐ cusp tion pro-p anab. + hidden endom. compat. of log-volumes w/ log-links invariant after admitting (Ind3) no interference by const. mult. rig. ell. cusp tion pro-p anab. + hidden endom. no interference by F mod O v = µ v arch. theory: Aut-hol. space ( ) ell. cusp tion invariant after admitting (Ind2) Ẑ -indet. only µ is involved multirad. protected from Ẑ -indet. by mono-theta cycl. rig. quadratic str. of Heisenberg gp protected from Ẑ -indet. by Ẑ Q >0 = {1} picture: permutable (étale after admitting (Ind1) ) (autom. of processions are included) 48 / 56

49 Some questions 49 / 56

50 How about the following variants of Θ -link? i) {q j2 } j q N (N > 1) ii) {(q j2 ) N } j q (N > 1) 50 / 56

51 i) {q j2 } j q N it works l ht deg 0 & deg l N 0 (ht) + (indet.) (as for N l) (When N > l the inequality is weak) 51 / 56

52 ii) {(q j2 ) N } j q it DOES NOT work! 52 / 56

53 Because 1 Θ Θ N mono-theta replace cycl. rig. mono-theta cycl. rig. comes from the quadraticity of [, ] cf. [EtTh, Rem2.19.2] Θ N (N > 1) / Kummer compat. 53 / 56

54 2 mono-theta constant multiple rig. as well ext n str. of 0 O ( ) Aut C (( )) Aut D (( )) bs ) 1 cf. [EtTh, Rem5.12.5] 54 / 56

55 3 vicious cycles Θ N zero of order = N > 1 at cusps various Frob-like µ Kummer theory étale-like µ cusp Kummer log Kummer cf. [IUT III, Rem.2.3.3(vi)] log Kummer loop one loop gives once N-power 55 / 56

56 IF it WORKED 0 N(ht) + (indet.) (ht) 1 N (1 + ε)(log-diff. + log-cond.) contradition to a lower bound given by analytic number theory (Masser, Stewart-Tijdeman) 56 / 56