F om Hadele–Hidele Sys ems o To ic F obenioids
A Func o ial T ans e o he To us–G aded Valua ion Skele on
Aleksanda Pe išić
Augus 2025
Abs ac
We gi e a clean way o pass om an analy ic Hadele–Hidele sys em (a uni a y model
o he adelic
ax+b
ac ion wi h a o us/Floque closu e o he log–scale) o a pu ely combi-
na o ial, o us–g aded F obenioid ha keeps only deg ees, he placemen o p ime powe s
on he
u
–ci cle, and he ci cle holonomy. In sho : we o ge addi ion and emembe he
alua ion/monod omy laye . We cons uc a small g aded ca ego y
TF
(
O
)and a canonical
unc o
T:O 7−→ (TF(O),deg,Lα),
unc o ial unde uni a y equi alences and deg ee–p ese ing in e wine s, compa ible wi h
Hecke wis s, and—o e unc ion ields wi h ci cle leng h
L
=
log q
—an equi alence wi h he
F obenioid o e ec i e di iso s. No Fou ie /Poisson iden i y is used; he ans e is ex ac ed
om ope a o s and o us holonomy.
One–page o e iew (plain language)
•
Wha a els. F om he analy ic side we keep: (i) he lis o p ime powe s
pk
(“ ee h”);
(ii) hei deg ee
klog p
; (iii) whe e each deg ee si s on he w apped log–scale
u∈R/LZ
;
and (i ) he ci cle’s holonomy eiα (Floque phase).
•
Wha does no a el. The addi i e/shi ope a o and any Poisson/ he a iden i ies.
Those ha e no unc o ial home in ou a ge .
•
The a ge objec . A small g aded ca ego y
TF
(
O
)whose objec s a e ini e mul ise s o
p ime powe s and whose mo phisms a e he deg ee- and o us–placemen –p ese ing maps
be ween hem; i ca ies a deg ee unc o and an S1local sys em eco ding monod omy.
•
Main heo em in one line. The e is a canonical, uni a y–in a ian unc o T om he
analy ic “ oo h/winding” ca ego y o
TF
; o e unc ion ields his iden i ies
TF
wi h he
usual F obenioid o e ec i e di iso s.
•
Why he o us? W apping he log–scale (
u∼u
+
L
) makes monod omy isible. O e
numbe ields his is non i ial; o e
K/Fq
(
T
)wi h
L
=
log q
i collapses o a i ial ci cle
bundle, e lec ing he amilia deg ee.
Toy example. Take
L
=
log
10. The objec [2
3
]has deg ee 3
log
2and si s a angle
θ
=
(3
log
2)
mod L
on he
u
–ci cle. A Hecke wis ac s by ephasing he ci cle local sys em bu
lea es deg ee and
θ
unchanged. O e a unc ion ield wi h
L
=
log q
,
θ≡
0 o all p ime powe s
(since log N(p)is a mul iple o log q).
1
Quick glossa y
Hadele/Hidele
H= Hamil onian/Heisenbe g/Hilbe . Hadele and Hidele a e he gene a o -
le el lows implemen ing he ax+bac ion, no he ings/g oups A,A×.
Tee h pkSpec al slices indexed by p ime powe s; deg ee klog p.
Winding ope a o wReco ds laps a ound he u–ci cle; W(α) = eiαwhas holonomy eiα.
Deg ee unc o Monoid homomo phism deg([pk]) = klog p.
To ic F obenioid
The ca ego y ha keeps only ee h, deg ee, ci cle placemen , and holonomy.
1 In oduc ion
A ecu ing heme in a i hme ic geome y is ha he mul iplica i e/ alua ion skele on is obus
(i su i es anabelian passage), whe eas he addi i e s uc u e is agile and o en delibe a ely
o go en. Analy ic models o he adelic
ax+b
ac ion e lec his spli : he e is a shi (addi i e)
side and a scale/p ime (mul iplica i e) side. Once he scale axis is gi en a o us (Floque )
closu e, he mul iplica i e side admi s a clean, unc o ial p ojec ion o a o us–g aded F obenioid,
ma ching he anabelian skele on.
Conc e ely, om a Hadele–Hidele sys em
O
we build a small g aded ca ego y
TF
(
O
)— he
o ic F obenioid—wi h p ime– oo h objec s [
pk
], deg ee
deg
([
pk
]) =
klog p
, and a ci cle local
sys em
Lα
wi h holonomy
eiα
inhe i ed om he Floque closu e. The p ojec ion is canonical and
unc o ial; in he unc ion ield case i becomes an equi alence wi h he F obenioid o e ec i e
di iso s. The poin is ha
TF
(
O
)can be compa ed di ec ly o he F obenioid o e ec i e
di iso s wi hou making any ze a- o he a- ype choices on he analy ic side.
Con ibu ions. We make he ollowing i ems p ecise.
(C1)
Axioma iza ion o Hadele–Hidele sys ems (De ini ion 2.1): addi i e gene a o
Hade
, mul-
iplica i e gene a o
Hide
=
N
+
Pp
(
log p
)
Np
, commu a o [
Hide, Hade
] =
iHade
, p ime
coun e s Np, and o us closu e (L, α)wi h winding ope a o w.
(C2)
Cons uc ion o he o ic F obenioid
TF
(
O
)wi h deg ee unc o
deg
and ci cle local sys em
Lα—no Fou ie /Poisson inpu .
(C3)
A canonical unc o T om Hadele–Hidele sys ems o o ic F obenioids (Theo em 5.2),
in a ian unde uni a y equi alence and na u al o deg ee–p ese ing in e wine s.
(C4)
Compa ibili y wi h a i hme ic idèle class cha ac e s (Hecke wis s) as o so ac ions on
TF (P oposi ion 6.1).
(C5)
O e unc ion ields wi h
L
=
log q
, an equi alence be ween
TF
(
O
)and he F obenioid o
e ec i e di iso s (Theo em 7.1).
Scope. We do no ecas non-abelian anabelian/IUT echniques analy ically; we isola e a
common o us–g aded laye . The addi i e ope a o and Poisson/ he a iden i ies ha e no
coun e pa in ou a ge ca ego y and a e explici ly le behind.
Rema k 1.1 (Naming: he “H” in Hadele/Hidele).The p e ix Hs ands o Hamil onian/Heisenbe g/Hilbe .
Ou Hadele and Hidele a e he ope a o -le el lows on a Hilbe space (gene a o s
Hade
,
Hide
),
no he adèles/idèles hemsel es. W i ing H-adele (Hadele) and H-idele (Hidele) lags his
iewpoin .
2
No a ion
We w i e
A
o adèles o
Q
,
A×
o idèles, and ix a ci cle leng h
L >
0and Floque phase
α∈R/
2
πZ
. The se Ω =
{klog p
:
pp ime, k ∈N}
is he p ime “ oo h” se along he log–scale.
2 Hadele–Hidele sys ems
We abs ac jus he mul iplica i e/ o us s uc u e we need.
De ini ion 2.1 (Hadele–Hidele sys em).AHadele–Hidele sys em Oconsis s o da a
(H, U, Hade, Hide,{Np}p, N, L, α, w, W(α)),
whe e:
(H1) H
is a complex Hilbe space and
U
: (
a,
)
∈A ⋊ A×7→ U
(
a,
)
∈ U
(
H
)is a uni a y
ep esen a ion o he adelic ax+bg oup. (Think: shi by aand scale by ac uni a ily.)
(H2) Hade
and
Hide
a e (essen ially sel -adjoin on a common co e) in ini esimal gene a o s o
a7→ U
(
a,
1) and
u7→ U
(0
, u
)wi h
| u|A
=
eu
.(Think: he shi and dila e Hamil onians.)
(H3)
Commu a o : [
Hide, Hade
] =
i Hade
on he co e. (Think: he usual dila ion– ansla ion
ela ion.)
(H4) P ime coun e s: The e a e commu ing numbe ope a o s Npwi h
Hide =N+X
p
(log p)Np,
whe e
N
gene a es he connec ed no m low. Fo each
p
and
k∈N
, he oo h
klog p∈
Ω
is an eigen alue o Np( he pk–band). (Think: Npcoun s how many p’s.)
(H5)
To us closu e: Fix
L >
0and
α∈R/
2
πZ
. The e is a uni a y
W
(
α
) =
eiαw
(Floque
holonomy) commu ing wi h
Hade, N
and all
Np
, whe e wis he in ege winding ope a o
implemen ing u∼u+L.(Think: w ap uon a ci cle and emembe he phase.)
Two sys ems a e uni a ily isomo phic i a uni a y in e wine p ese es hese s uc u es.
Rema k 2.2. We e ain only he mul iplica i e/ o us laye o be ans e ed. No Fou ie /Poisson
iden i y o he a s a e is assumed.
3 The o us and winding
Le
S1
u
=
R/LZ
be he ci cle ob ained by w apping he log–scale
u
. The ope a o w om
(H(H5)) has spec um in
Z
and eco ds winding numbe ; he uni a y
W
(
α
)has holonomy
eiα
along a lap. The pai (
S1
u,Lα
)unde lies he o ic g ading, wi h
Lα
he
S1
–local sys em o
holonomy eiα.
4 The o ic F obenioid
We now de ine he ca ego ical a ge .
De ini ion 4.1 (To ic F obenioid).Gi en O, de ine a small ca ego y TF(O)as ollows.
•
Objec s: ini e o mal sums
X
=
Pcp,k
[
pk
]wi h
cp,k ∈N
.(Think: mul ise s o p ime
powe s.)
3
•
Mo phisms: gene a ed by pa ial isome ies [
pk
]
→
[
pk
]and inclusions [
pk
]
⊕
[
pk′
]
→
[pk+k′], subjec o p ese ing:
(M1) deg ee: deg([pk]) = klog pex ends addi i ely;
(M2) o us posi ion: θ([pk]) = (klog p) mod L∈S1
u;
(M3) winding: he winding a ached o klaps equals k(compa ible wi h w).
•Deg ee unc o : deg : TF(O)→(R≥0,+).
•Local sys em: a ci cle local sys em Lαon S1
u, pulled back o objec s ia θ.
We w i e (TF(O),deg,Lα) o his o ic F obenioid.
Rema k 4.2. This is he alua ion/deg ee skele on wi h explici monod omy bookkeeping; i
o ge s addi ion.
5 The ans e unc o
We i s isola e a small ∗–ca ego y ex ac ed om O.
De ini ion 5.1 (Too h/winding
∗
-ca ego y).Le C(
O
)be he s ic
∗
–ca ego y whose objec s
a e ini e o hogonal sums o spec al subspaces o he
Np
indexed by ee h [
pk
], and whose
mo phisms a e bounded ope a o s gene a ed by pa ial isome ies ha commu e wi h
N
and w
and p ese e oo h labels.
Theo em 5.2 (Func o ial ans e ).The e is a canonical unc o
T:C(O)−→ TF(O)
ha on objec s sends a spec al subspace o oo h [
pk
] o he symbol [
pk
], and on mo phisms
sends any deg ee/ o us–p ese ing pa ial isome y o he co esponding gene a o in
TF
. The
unc o is:
(F1) well-de ined (independen o ep esen a i es);
(F2) unc o ial in O(uni a y isomo phisms induce isomo phisms o o ic F obenioids);
(F3) essen ially su jec i e on objec s and ull on deg ee–p ese ing mo phisms;
(F4) compa ible wi h he deg ee unc o and he pullback o he ci cle local sys em Lα.
P oo ske ch.
(F1) Spec al subspaces o
Np
decompose
H
in o sums labeled by [
pk
]; sending
hese o symbols is canonical. (F2) Uni a y in e wine s p ese e spec a o
Np
,
N
, and w.
(F3) Any ini e sum o symbols is ealized by spec al subspaces; ullness holds because any
combina o ial deg ee/ o us–p ese ing map is implemen ed by a pa ial isome y suppo ed on
he slices. (F4) Compa ibili y wi h
deg
ollows om
Hide
=
N
+
P
(
log p
)
Np
; compa ibili y wi h
Lαuses commu a ion wi h W(α) = eiαw.
Rema k 5.3 (Wha is no ans e ed).No use is made o
Hade
beyond he commu a o axiom;
Poisson/ he a iden i ies and analy ic eadou s a e in en ionally absen om he a ge . The
unc o Tis a p ojec ion on o he o us–g aded alua ion laye .
4
6 A i hme ic wis s as o so ac ions
A i hme ic idèle class cha ac e s ac na u ally on he o ic F obenioid by phase o so s.
P oposi ion 6.1 (Hecke compa ibili y).Le
ω
:
CQ
=
A×/Q×→S1
be a uni a y idèle class
cha ac e . Then
ω
de ines an au oequi alence o
TF
(
O
) ha ixes objec s and enso s he local
sys em along S1
uby a cons an phase, p ese ing deg and θ. This is unc o ial:
(O, ω)7−→ (TF(O),deg,Lα⊗ω)
canonically as g aded ca ego ies wi h local sys em.
Idea.
A Hecke cha ac e es ic s o phases on p ime powe s compa ible wi h mul iplica i i y.
Since
TF
o ge s addi ion and keeps only deg ee and o us placemen , he e ec is o ephase
he ci cle local sys em by a cons an S1 wis , lea ing deg and θunchanged.
7 Func ion ield case: an equi alence
The unc ion ield se ing a o ds an exac iden i ica ion.
Theo em 7.1 (Equi alence o e unc ion ields).Le
K/Fq
(
T
)be a global unc ion ield and
choose L= log q. Then o any p ime ideal pand k≥1,
θ([pk]) ≡0 (mod L),deg([pk]) = kdeg(p) log q.
The unc o Tinduces an equi alence
TF(O)≃Di e (K),
whe e
Di e
(
K
)is he F obenioid o e ec i e di iso s wi h
deg
([
pk
]) =
kdeg
(
p
)
log q
and i ial
ci cle local sys em.
P oo ske ch.
Since N(
p
) =
qdeg p
, we ha e
log
N(
p
) =
deg p·log q
, so (
klog
N(
p
))
mod log q≡
0.
Thus o us placemen is cons an , and he emaining g ading is he amilia in ege deg ee
(scaled by log q). Objec s and mo phisms hen ma ch hose o Di e (K).
8 Func o iali y and s abili y
P oposi ion 8.1 (Uni a y in a iance).I
O
and
O′
a e uni a ily isomo phic Hadele–Hidele
sys ems, hen TF(O)∼
=TF(O′)canonically as g aded ca ego ies wi h local sys em.
Ske ch.
A uni a y in e wine p ese es spec a o
Np
,
N
, and w, hence p ese es
deg
,
θ
, and
he ci cle local sys em.
P oposi ion 8.2 (S abili y unde deg ee–p ese ing in e wine s).Le
T
:
O → O′
be a bounded
in e wine commu ing wi h
N
, all
Np
, and
W
(
α
). Then Tmaps
T
o a mo phism o o ic
F obenioids p ese ing deg and Lα.
Ske ch.
Such
T
espec s he oo h decomposi ion and o us holonomy, hence descends o he
combina o ial da a o TF.
5
9 Limi a ions and ou look
The p ojec ion Tis in en ionally blind o addi ion: i canno see
Hade
beyond i s commu a o
wi h Hide, and i uses no Poisson/ he a iden i y. Two di ec ions emain:
•
Tannakian li . Enhance
TF
o a neu al Tannakian ca ego y whose undamen al g oup
eco e s (a leas ) he abelianized Galois g oup ia class ield heo y; a i hme ic wis s ac
as enso au oequi alences.
•
Non-abelian laye . Ex ending beyond alua ions would need ca ego ical s uc u es encoding
empe ed undamen al g oups; ou analy ic da a alone a e insu icien .
•
Expanding lows. One may adjoin u he one–pa ame e lows wi h p esc ibed commu a o s
wi h
Hade
,
Hide
and
W
(
α
). When such lows p ese e
N
,
Np
and w, hey pass h ough T;
o he wise hey emain analy ic deco a ions ha he p ojec ion o ge s.
10 Connec ions wo h a casual look
The ollowing links seem na u ally compa able o he o us–g aded alua ion laye (
deg
, placemen
modL, holonomy):
•
Dynamical ze a (Ruelle/Selbe g). P imi i e pe iodic o bi s
↔
p ime powe s; ans e
ope a o s and wis ed Eule p oduc s.
•
Iha a ze a and g aphs. P imi i e cycles
↔
p ime powe s; amilies o weigh ed g aphs
wi h cycle leng hs app oxima ing klog p(mod L).
•
Noncommu a i e geome y/KMS (Bos –Connes). Time e olu ion om he mul i-
plica i e gene a o on a semig oup
C∗
-algeb a; KMS phases con olled by o us–g aded
da a.
•
The modynamic o malism. Weigh s on ee h as a po en ial; opological p essu e s.
log Eule p oduc s.
•
T opical/Be ko ich/A akelo skele ons. Package as a me ized g aph plus an
S1
local sys em; check unc o iali y unde pullbacks.
•
SYZ–s yle mi o oys. The
u
–ci cle as base, p ime ee h as singula ibe s; Landau–
Ginzbu g monod omy.
•
Reno maliza ion Hop algeb as. P imes as p imi i es and lows as cha ac e s; compa e
associa ed g adeds wi h he o ic F obenioid.
11 A mic o example: he p ime 3
We close wi h a conc e e h ee-s ep example showing how a single p ime powe a els om he
analy ic Hadele–Hidele side o he o ic F obenioid, and how i looks in he unc ion ield case.
S ep 1: analy ic side (a single oo h)
Fix a Hadele–Hidele sys em Owi h ci cle leng h L > 0. Fo he p ime p= 3, le
H3k⊂ H
be a nonze o spec al subspace on which:
6
•N3ac s by he eigen alue k;
•Npac s by 0 o all p= 3;
•Nac s wi h some spec um con ained in an in e al Ik⊂R.
On H3k he mul iplica i e gene a o es ic s o
Hide
H3k=N
H3k+klog 3 ·id,
so he oo h [3
k
]has deg ee
klog
3in he sense o ou cons uc ion. W apping he log–scale
u
on
S1
u=R/LZ, his oo h si s a ci cle angle
θk:= (klog 3) mod L∈S1
u,
and he winding ope a o w eco ds how many laps a ound
S1
u
he low has made. The Floque
uni a y W(α) = eiαwac s on H3kby a phase de e mined by he winding numbe .
S ep 2: image in he o ic F obenioid
Applying he unc o To Theo em 5.2, he spec al subspace H3kis sen o he objec
[3k]∈TF(O).
On his objec we e ain exac ly:
• he deg ee
deg([3k]) = klog 3;
• he o us posi ion
θ([3k]) = θk= (klog 3) mod L∈S1
u;
• he ibe o he ci cle local sys em Lαa θk, wi h holonomy eiα a ound he ci cle.
Any pa ial isome y
V:H3k−→ H3k
ha commu es wi h
N
and wand p ese es he oo h label [3
k
]is mapped by T o a mo phism
T(V) : [3k]−→ [3k]
in
TF
(
O
), i.e. o an endomo phism o he co esponding objec ha espec s deg ee and o us
placemen .
I L= log 10, o example, hen
deg([32]) = 2 log 3, θ([32]) = (2 log 3) mod log 10,
so he oo h [32]is a poin on he u–ci cle a ha angle, ca ying he local sys em Lα.
7
S ep 3: unc ion ield shadow
Now specialize o he unc ion ield si ua ion o Theo em 7.1. Take a global unc ion ield
K/Fq
(
T
)wi h
q
= 3 and choose
L
=
log q
=
log
3. Fo any p ime ideal
p⊂ OK
o deg ee
deg p
,
we ha e
log N(p) = deg p·log 3,
so
θ([pk]) = (klog N(p)) mod log 3 ≡0 (mod log 3).
In pa icula , he ci cle coo dina e o e e y oo h collapses o 0, and he only emaining da a
a e he in ege deg ees kdeg(p).
Unde he equi alence
TF(O)≃Di e (K)
om Theo em 7.1, ou analy ic oo h [3
k
]is seen only h ough i s deg ee: i co esponds o an
e ec i e di iso o he app op ia e deg ee (up o he usual iden i ica ion o p imes wi h places
o
K
). The o ic e inemen ( he angle
θk
and he holonomy o
Lα
) is in isible in he unc ion
ield F obenioid, which ma ches he usual pic u e: o e unc ion ields, he alua ion skele on is
pu ely disc e e.
This iny example illus a es he gene al philosophy:
On he analy ic side, a oo h such as [3
k
]li es on a o us wi h a non i ial local
sys em; he unc o Tkeeps exac ly he deg ee, he ci cle posi ion, and he holonomy.
In he unc ion ield limi wi h
L
=
log q
, he ci cle collapses and only he disc e e
F obenioid o e ec i e di iso s su i es.
Acknowledgmen s
Discussions a ound he addi i e/mul iplica i e spli and o us closu es inspi ed his o maliza ion;
any emaining imp ecision is he au ho ’s.
Re e ences
[1]
J. T. Ta e, Fou ie Analysis in Numbe Fields and Hecke’s Ze a-Func ions, Ph.D. hesis,
P ince on (1950); ep in ed in Cassels–F öhlich, Algeb aic Numbe Theo y, Academic P ess
(1967).
[2] A. Weil, Basic Numbe Theo y, Second Edi ion, Sp inge (1974).
[3] S. Mochizuki, The Geome y o F obenioids, Publ. RIMS Kyo o Uni . 37 (2001), 163–223.
[4]
S. Mochizuki, Topics in Absolu e Anabelian Geome y, Publ. RIMS Kyo o Uni . 40 (2004),
1–40.
8
