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

Transcription

1 [IUTch-III-IV] from the Point of View of Mono-anabelian Transport IV Main Theorem and Application Yuichiro Hoshi RIMS, Kyoto University July 22, 2016 Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

2 41 Kummer-compatible Multiradial Representations 42 Log-Volume Estimates 43 Diophantine Inequalities Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

3 41 Kummer-compatible Multiradial Representations In summary, the following hold: 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] Recall: HT D-Θ±ell NF Prc( D T ) def = ( { D 0 } { D 0, D 1 } { D 0,, D l } ) Prc( D T ) def = ( { D 0} { D 0, D 1} { D 0,, D l }) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

4 Rét : the collection of data consisting of: (aét ) V v v Q V Q, 0 j l Prc( D T ) I( S± j+1 ; D V Q ) I Q ( S± j+1 ; D V Q ), I( S± j+1,j; D v ) I Q ( S± j+1,j; D v ) [with the procession-normalized mono-analytic log-volumes] (bét ) v V bad HT D-Θ±ell NF Ψ LGP ( HT D-Θ±ell NF ) v in j J IQ ( S± j+1,j; D v ) (cét ) 1 j l HT D-Θ±ell NF ( M D MOD ) j = ( M D mod) j in I Q ( S± j+1 ; D V Q ) ( F D MOD ) j ( F D mod) j [whose ass degree may be com d by means of the log-vol of (aét )] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

5 log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log Θ µ LGP log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log Prc( D T ) Prc( D T ) (aét ) (aét ) Moreover: Rét up to (Ind1) Rét up to (Ind1), where up to (Ind1) def up to indeterminacies induced by Aut(Prc(D T )) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

6 étale-like holomorphic étale-like holomorphic theta values (bét ) number fields (cét ) étale-like mono-analytic tensor packet log-shells (aét ) (Ind1) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

7 m Z R Frob : the collection of data consisting of: (a Frob ) V v v Q V Q, 0 j l I( S± j+1 ;m F µ V Q ) I Q, I( S± j+1,j;m Fv µ ) I Q [with the procession-normalized mono-analytic log-volumes] (b Frob ) v V bad Ψ F LGP ( m HT Θ±ell NF ) v in j J IQ ( S± j+1,j;m F µ v ) (c Frob ) 1 j l ( m M MOD ) j = ( m M mod) j in I Q ( S± j+1 ;m F µ V Q ) ( m F MOD ) j ( m F mod) j [whose ass degree may be com by means of the log-vol of (a Frob )] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

8 Then: (a Kmm ) (a Frob ) up to (Ind2) (aét ) up to (Ind2), compatible w/ the respective log-volumes, where up to (Ind2) def up to indeterminacies induced by Ism v, {±1} [cf 13] (b Kmm ) (b Frob ) (bét ) (c Kmm ) (c Frob ) (cét ) m C LGP CLGP D, m Clgp Clgp D Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

9 Moreover: Ψ F LGP ( m HT Θ±ell NF ) Ψ LGP ( HT D-Θ±ell NF ) of (b Kmm ) ( m M MOD ) j ( M D MOD ) j of (c Kmm ) are mutually compatible with one another [wrt m] [cf the non-interference property discussed in 23 and 33] m C LGP C D LGP of (ckmm ) is mutually compatible with one another [wrt m] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

10 Recall: The upper semi-compatiblity discussed in 15: v V non ( (Ψcns ( m F ) j ) v ) Gal I( S± j+1 ;m F vq ) Kmm log I( S± j+1 ;m+1 F vq ) Kmm log I( S± j+1 ; D vq ) I( S± j+1 ; D vq ) v V arc a similar diagram One may introduce the following indeterminacy (Ind3) to (a Kmm ): (Ind3): As one varies m, (a Kmm ) are [not precisely compatible but] upper semi-compatible Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

11 log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log Θ µ LGP log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log 0 F µ 0 F µ is compatible, under (Ind1, 2, 3), w/: F µ ( D ) F µ ( D ) F µ env ( D > ) F µ env ( D > ) R tht R tht, as well as the Gal eval R tht (bét ) [cf 31] R NF R NF, as well as the Gal eval R NF (cét ) [cf 21] relative to the Kmm poly-isom and poly-isom of mono-theta env Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

12 Fr-like holomorphic Fr-like holomorphic theta values (b Frob ) number fields (c Frob ) (b Kmm ) (c Kmm ) étale-like holomorphic étale-like holomorphic theta values (bét ) number fields (cét ) mono-analytic tensor packet log-shells (aét ) (Ind1, 2, 3) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

13 42 Log-Volume Estimates log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log Θ µ LGP log 1 HT Θ±ell NF log 0 HT Θ±ell NF log 1 HT Θ±ell NF log Let us start with a q-plt obj 0 q 0 C at 0 and compute deg(0 q) By definition, 0 C LGP 0 C of Θ µ LGP maps the isom class [0 Θ] of a Θ-pilot object 0 Θ 0 C LGP at 0 to the isom class [0 q] of 0 q Let us observe that deg( 0 q) (resp deg( 0 Θ)) can be computed by means of the log-volume of 0 q (resp 0 Θ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

14 Let us apply the multiradial representation of R Frob [cf 41] 0 F µ Θ µ LGP 0 F µ LGP multiradiality (Ind1, 2, 3) 0 F µ LGP ; [0 q] [ 0 Θ] [ 0 Θ] In other words, roughly speaking, the multiradial representation asserts that [ 0 q] can be interpreted as [ 0 Θ] up to (Ind1, 2, 3) The log-volume on the log-shells is invariant with respect to the indeterminacies (Ind1) and (Ind2) On the other hand, by the upper semi-compatibility indeterminacy (Ind3), ie, commutativity at the level of inclusions of regions in log-shells, we must consider the log-volume of an object up to (Ind3) as an upper bound Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

15 In order to utilize the above interpretation [ie, not 0 q = 0 (Ind1, 2, 3) Θ but [ 0 q] = (Ind1, 2, 3) [ 0 Θ]] or, more precisely, to compute an upper bound of the log-volume of [not 0 Θ but] 0 Θ relative to the pt of view of [not 0 but] 0, it is nece y to work w/ the coll n of modules, ie, holomorphic hull In summary, we conclude: deg(q) (= the log-volume of the q-pilot object) the log-volume of the holomorphic hull of (Ind1, 2, 3) Θ-pilot object Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

16 43 Diophantine Inequalities log-vol of q-pilot log-vol of holo hull of (Ind1, 2, 3) Θ-pilot Rough Estimate at v V bad : µ log (q v O Kv ) 1 l µlog q j2 v q (l ) 2 v q v (I v ) 0 (I v ) 1 (I v ) 0 (I v ) j (I v ) 0 (I v ) l Recall: ord(q j2 v ) = j2 2l ord(q v), 0 < ord(q v ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

17 l = l/2 l j=1 j + 1 = (l/2)2 /2 l j=1 j2 = (l/2) 3 /3 µ log ((I v ) 0 (I v ) j ) (j + 1) ord(d Kv /Q p ) ord(q v ) 1 l l j=1 1 ( (j + 1) ord(dkv/q p ) j2 2l ord(q v) ) 2 = ( (l/2) 2 ord(d l 2 Kv /Q p ) (l/2)3 ord(q 2l 3 v) ) l = (ord(d 4 Kv /Q p ) 1 ord(q 6 v) ) l = (ord(d 4 Kv /Q p ) 1 12 (1 ) ord(q 6 l 2 v ) ) ord(q v ) l 4 ord(q v ) (ord(d Kv /Q p ) 1 12 (1 6 l ) ord(q v) ) ord(q v) (1 12 l 2 ) 1 ord(d Kv/Q p ) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

18 P def = P 1 Q D def = {0, 1, } red, U P def = P \ D For λ U P (Q) = Q \ {0, 1}: A λ : the elliptic curve/q(λ) def d by y 2 = x(x 1)(x λ) F λ def = Q(λ, 1, A λ [3 5](Q)) E λ def = A λ Q(λ) F λ has at most split multipl red at V(F λ ) q λ ADiv(F λ ): the eff arith div det d by the q-param r of E λ /F λ f λ ADiv(F λ ): the eff reduced arithmetic divisor det d by q λ d λ ADiv(F λ ): the eff arith div det d by the different of F λ /Q def d λ = [Q(λ) : Q] d def λ = d λ Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

19 For a prime number l 5: (λ, l): admissible def an initial Θ-data (Q/F λ, E λ \ {o}, l, ) st E λ has good reduction at v V(F λ ) good of residue char 2l Then log-vol of q-pil log-vol of holo hull of (Ind1, 2, 3) Θ-pil (λ, l): admissible 1 6 deg(qbad λ ) (1 + 80d λ ) (deg(d λ ) + deg(f λ )) + 20 (d λ l + η prm ), l where η prm R >0 st {prime numbers η} 4 η ( η > η 3 log(η) prm) [cf the above Rough Estimate ] Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

20 By means of this, let us prove the following ( ): ( ): d Z 1 ϵ R >0 K P (C): an ι-stable cpt domain S: a finite set of prime numbers st 2 S K p P (Q p ): a certain compact domain [p S] K def = P (Q) K p S K p, ie, a compactly bounded subset d def The function on K = { λ K d λ d } given by λ 1 6 deg(q λ) (1 + ϵ) (deg(d λ ) + deg(f λ )) is bounded above Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

21 Proof of ( ): By applying the prime number theorem, the theory of arithmeticity of elliptic curves, the finiteness of { λ U P (Q) d λ d, deg(q λ ) C } for C R, the theory of Galois actions on torsion points of elliptic curve, one may obtain the foll g: For all but finitely many λ K d, l λ st (a) (λ, l λ ): admissible (In particular: 1 (b) 6 deg(qbad λ ) (1 + 80d λ l λ ) (deg(d λ ) + deg(f λ )) + 20 (d λ l λ + η prm )) (c) ord lλ (q l ) < deg(q λ ) 1/2, where V(F λ ) l l λ (d) deg(q λ ) 1/2 l λ 10 d deg(q λ ) 1/2 log(2 d deg(q λ )), where d def = d Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

22 (a), (c) an upper bound of the function λ 1 6 deg(q λ) 1 6 deg(qbad λ ) deg(q λ ) 1/2 log(2d deg(q λ )) (b), (d) 1 6 deg(qbad λ ) (1 + d deg(q λ )(deg(d ) 1/2 λ ) + deg(f λ )) an upper bound of the function λ +200(d ) 2 deg(q λ ) 1/2 log(2d deg(q λ )) + 20η prm (1 2 (60d ) 2 log(2d deg(q λ )) 5 deg(q λ ) 1 ) 1/2 6 deg(q d λ) (1 + deg(q λ )(deg(d ) 1/2 λ ) + deg(f λ )) Thus, an upper bound of the function λ 1 6 deg(q λ) (1 + ϵ)(deg(d λ ) + deg(f λ )) Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23

23 By the above ( ), together with the theory of noncritical Belyi maps, we obtain the following theorem: d Z 1 ϵ R >0 L: a number field V : a projective smooth curve /L D V : a [possibly empty] reduced divisor st Ω 1 V/L (D): ample d def The function on (V \ D) = { x V \ D [κ(x) : Q] d } given by λ ht Ω 1 V /L (D) (1 + ϵ) (log-diff V + log-cond D ) is bounded above Yuichiro Hoshi (RIMS, Kyoto University) IUTch-III-IV July 22, / 23