The universal thickening of real & complex numbers:
towards an archimedean analogue of Fontaine’s theory
Caterina Consani – Johns Hopkins U.
(project in collaboration with Alain Connes)
Texas A&M – May 2, 2014
MAIN GOAL: Define an ∞-adic theory of periods in direct analogy
with p-adic Hodge theory
Aut(R) = {1},
Theorem 1
AutR(C) = Z/2Z
There exist universal field covers:
θ
W (R♭ ) R,
θ
W (C♭ ) C
with larger groups of automorphisms ({Fλ}λ∈R× ∈ Aut(W (R♭)))
+
Theorem 2 The infinite dimensional real, resp. complex vector spaces
Ω := Ker(θ)/Ker(θ2)
contain the independent set of periods: {πp := [p] − p | p ∈ Z prime}
resp. the independent complex periods:
{πp := [e(log p)] − p | p ∈ Z prime},
ω := ([e(2πi)] − 1)/([e(πi)] − 1)
I. Review of p-adic Hodge theory
L = number field,
X/L = proper, smooth alg. variety
i (X ) ≃ C ⊗ H i (X, Z)
C ⊗L HdR
Z B
/L
Goal of the theory of p-adic periods:
K = finite extension of Qp,
GK := Gal(K̄/K)
X/K = proper, smooth alg. variety
construct analogous isomorphisms:
(HT/dR conjectures)
i
i (X
(??) ⊗K HdR
/K ) ≃ (??) ⊗Qp Hét (XK̄ , Qp )
Fontaine’s strategy: Introduce rings of periods B (Qp-algebras)
with an action of GK and additional structures so that:
DB(Vp) := (B ⊗Qp Vp)GK
BGK − module
becomes an invariant of any “good” (B-admissible) GK -module Vp:
B ⊗BGK DB(Vp) ≃ B ⊗Qp Vp
B = BdR
(filtered) field of p-adic periods,
BGK = K
i (X , Q )
X/K = smooth, proper variety: Vp := Hét
p
K̄
i (X , Q )) ≃ H i (X
DdR(Hét
p
/K )
K̄
dR
(Faltings)
(filtered) K-isomorphism
i
i (X
BdR ⊗K HdR
/K ) ≃ BdR ⊗Qp Hét(XK̄ , Qp)
B+
dR Cp : the universal Galois-equivariant cover
3 fundamental steps (K = Qp):
(1) Universal perfection C♭p of the p-perfect field Cp := (Q̄p)∧
(2) p-isotypical Witt construction: C♭p ⊃ OC♭
p
W (OC♭ )
p
1]
)[
(3) B+
is
a
suitable
completion
of
W
(O
♭
dR
p
C
p
C♭p := {x := (x(n))n≥0| x(n) ∈ Cp; (x(n+1))p = x(n)}
m
(n+m)
(n+m)
p
(n)
(x + y)
:= lim (x
+y
) ,
m→∞
(xy)(n) := x(n)y (n)
perfect, complete field of characteristic p
∼
OC♭ := lim
←−pOCp −→ lim
←−pOCp /pOCp ,
p
x7→x
v7→v
x 7→ x̄ := (x(n)mod. p)n≥0
perfect, associative, commutative Fp-algebra
∑
1
b,+
B
:= W (OC♭ )[ p ] = {
[xk ]pk ∈ W (C♭p) ⊗ Qp| xk ∈ OC♭ }
p
p
k≫−∞
Zp-algebra (not complete dvr)
θ : W (OC♭ ) OCp
p
lift of
Fix:
∑
k≥0
θ0 : OC♭p OCp /(p),
ε = (ε(n) ∈ µpn (K̄))n≥0 ∈ Zp(1)
ω := ([ε] − 1)/([ε1] − 1),
!!
[xk ]pk →
7
ker(θ) = (ω) = ([p̃] − p)
∑ (0)
xk pk
k≥0
x 7→ x(0)
[ε(0) = 1, ε(1) ̸= 1, (ε(n+1))p = ε(n)]
[ε1] := [(ε(1), . . .)],
p̃(0) := p,
p
ε1 = ε
p̃ := (p̃(n))n≥0
∞
∑
+
b,+ /(ker(θ))n) = {f =
n|x ∈
BdR := lim
(
B
x
ω
n
n
←−
n=0
n∈N
Qp-algebra; complete dvr; with residue field Cp
Bb,+}
θ(1 − [ε]) = 0 =⇒ 1 − [ε] is “small” for the topology of B+
dR
t := log([ε]) =
∞
∑
([ε] − 1)
n−1
(−1)
n
n=1
Zp(1) ↩→ B+
dR ,
g ∈ Gal(Q̄p/Qp) :
n
ε 7→ t
∈ B+
dR
GQp -equivariant
g(t) = g(log([ε])) = log([εχ(g) ]) = χ(g)t
t is a period of the cyclotomic character
GQ
BdR := B+
dR [1/t];
F iliBdR := tiB+
dR ;
Caveat:
φ(
∑
grBdR ≃
⊕
BdR p = Qp
Cp(i) =: BHT
i∈Z
(GQp -equivariant)
NO natural Frobenius structure on B+
dR (or on BdR)
k
k [xk ]p ) =
∑
p k
k [xk ]p
on Bb,+
BUT: φ([p̃] − p) = [p̃p] − p ∈
/ ker(θ)
II.
Fix:
R
κ∈Q
R♭
Perfection to characteristic one
|κ|∞ < 1,
|κ|2 = 1
R♭ := {x = (xn)n≥0 | xn ∈ R; (xn+1)κ = xn}
(∗)
m
κ
(x + y)n := lim (xn+m + yn+m) , (xy)n := xnyn
m→∞
THEOREM (1)
∼
R♭ −→ R,
x 7→ x0
bijection of multiplicative monoids.
(2)
−(n+m)
−(n+m) κm
κ
κ
′
(x + y)n = lim (x0
+ y0
)
m→∞
on R ∼ R♭
is not associative.
(3) The addition ⌣ defined by taking the limit of the graphs
Gκ,m in (∗), is multivalued and associative
Gκ= 1 ,m=1,n=0
1
(x, y) := (x, y, (x3 + y 3) 3 )
3
At m = 1 (cf. above):
Gκ= 1 ,m(x, y) is a univalent map
3
lim G
m→∞ κ= 1
3 ,m
(x, y) fails to be univalent on the line y = −x:
At m ≫ 0 (cf.above):
Gκ= 1 ,m
3
1
m
m
(x, y) := (x, y, (x3 +y 3 ) 3m )
x +′ y can be expressed in terms of x0 ∈ R ∼ R♭:
−(n+m)
x +′ y := ((x +′ y)n)n≥0 = ( lim (xκ
0
m→∞
x +′ y =


 x, if |x| > |y| or x = y;
y, if |x| < |y| or x = y;

 0, if y = −x
In particular:
However:
+ y0κ
x +′ x = x
−(n+m)
m
)κ )n≥0
|x| := |x0|
∀x ∈ R♭
the associativity wrt +′ fails
(y +′ x) +′ (−x) = x +′ (−x) = 0 ̸= y +′ (x +′ (−x)) = y +′ 0 = y
∀y ∈ R♭, |y| < |x| or y = x
Hyperfield structure on R♭ ∼ R


 x,
if |x| > |y| or x = y;
x ⌣ y := y,
if |x| < |y| or x = y;

 [-x,x], if y = −x
.x
.
. y
.0
. .x .y
.0
.x . .y
.0
R♭ = (R, ⌣, ·)
x, y ∈ R
.x ⌣ y
. .
.0
.x ⌣ y
.
.0
y
..x ⌣
. .
.0
(Viro) real tropical hyperfield
What is a hyperfield (F, +, ·) ?
(M. Krasner 1957-58)
Sum:
+ : F × F → P(F )∗,
(F, +) = canonical hypergroup
− x + y = y + x, ∀ x, y ∈ F ;
∃! 0,
0+x=x=x+0
− (x + y) + z = {∪(α + z) | α ∈ x + y} = x + (y + z)
− For x ∈ F,
∃! y ∈ F :
− (reversibility):
Product:
0∈x+y
(y := −x)
x ∈ y + z =⇒ z ∈ x − y
· : F × F → F,
(F ∗ = F \ {0}, ·) = group, 1F ̸= 0F
• the distributivity laws hold
Examples of hyperfields
1) K = {0, 1},
1 + 1 = {0, 1}
2) S = {−1, 0, 1},
1 + 1 = 1, − 1 − 1 = −1
1 − 1 = {−1, 0, 1},
sign : Z −→ S;
hyperfield of signs
| · | : S −→ K;
Spec (S)
Krasner hyperfield
F1 := (K, ·),
q
qqq
q
q
q
∃!qqqqq
q
qq
qqq
q
q
x q
q
Spec (R♭ ) o
o
B := S+ = {0, 1}
real tropical hypergeometry
Spec (Z)
Spec (B)
qq
qqq
q
q
qq
qqq
q
q
q
qqq
q
q
x q
q
Spec (F1 )
o
Spec (Rmax )
o
tropical geometry
Why NCG connects with hyperring theory (R, +, ·)?
- (R, +) canonical hypergroup
- (R, ·) multiplicative monoid with 1 ̸= 0
- distributivity laws hold.
Fact: R = commutative ring, G ⊂ R× subgroup. The set R/G with:
x̄ + ȳ := (xG + yG)/G
x̄ · ȳ := xyG
∀x̄, ȳ ∈ R/G
(coset)
is a hyperring
Theorem R = commutative ring, {1} ̸= G ⊂ R× subgroup
(Krasner) {0, 1} = K ⊂ R/G ⇐⇒ (G ∪ {0}) ⊂ R
Main example:
subfield
K ⊂ AQ/Q× adèle class space as K-hyperalgebra
The hyperfield R♭: main properties
(1) R♭ has characteristic one
(i.e.
x + x = x, ∀x ∈ R♭)
∼
♭ ), [φ(λ)](x) := sign(x)|x|λ group isom
(2) φ : R×
−→
Aut(
R
+
if λ ∈ Q×
+ , |λ|2 = 1:
(3) R♭ is perfect
[φ(λ)](x) = xλ
∼
(i.e. R♭ −→ R♭, x 7→ xn, ∀n = 2Z + 1)
(4) O := [−1, 1] ⊂ R♭
maximal compact sub-hyperring
(5) R♭ is complete for d(x, y) := |x0 − y0| on R♭
∼
R♭ → R, x 7→ x0
isometry for d
III. Witt construction for hyperfields
H = hyperfield
Definition (1) {W (H), ρ, τ } = Witt-model (W-model) of H:
W (H) = field,
ρ : W (H) −→ H hyperfield homomorphism
τ : H −→ W (H) hom of multiplicative monoids: ρ ◦ τ = idH
α
(2) {W (H)1, ρ1, τ1} −→ {W (H)2, ρ2, τ2} morphism of W -models:
α : W (H)1 → W (H)2
field homom:
τ2 = α ◦ τ1, ρ1 = ρ2 ◦ α
(3) {W (H)u, ρu, τu} universal W -model of H if ∀{W (H), ρ, τ } ∃!
α
u
{W (H)u, ρu, τu} −→
{W (H), ρ, τ }
(morphism of W -models)
Universal W -models are unique up-to unique isomorphism
The universal W -model of R♭
THEOREM (1) W (R♭) := {Frac(Q[R×
+]), ρu , τu = [·]}
universal W -model
×
−→
Q
[
R
R×
+ ],
+
x 7→ [x] ∈ Q[R×
+]
[·]
τu : R♭ ∼ R −→ W (R♭),
ρu
♭
W (R ) −→
R♭ ∼ R,
x0 := sup{xi},
(canonical generator)
[0] = 0, τu(−x) = −[x]
( )
∑
a0 x0
ai [xi ]
i
∑
ρu( b [y ] ) := sign
j j j
b0 y0
y0 := sup{yj },
ai, bj ∈ Q
(2) ρu induces an isomorphism of hyperfields:
∼
W (R♭)/G −→ R♭,
×
× → R♭ )
G := Ker(ρu : Frac(Q[R×
])
+
From:
♭ ),
R×
−→
Aut(
R
+ ∼
φ
λ 7→ [φ(λ)](x) = sign(x)|x|λ
It follows (for each fixed λ):
−1 ◦ ρ , τ ◦ φ(λ)}
]),
φ(λ)
{Frac(Q[R×
u u
+
ρu
])
−→
Frac(Q[R×
+
φ(λ)−1
♭
R −→
(universality ⇒) for each fixed λ:
τu ◦ φ(λ) = Fλ ◦ τu;
Corollary
R♭ ;
∃!
is also a W -model of R♭
φ(λ)
u
R♭ −→ R♭ −→
Frac(Q[R×
+ ])
τ
Fλ
×
Frac(Q[R+]) −→ Frac(Q[R×
+])
∼
ρu = φ(λ)−1 ◦ ρu ◦ Fλ
Aut(R♭) −→ Aut(W (R♭)),
φ(λ) 7→ Fλ
{Fλ}λ∈R× ∈ Aut(W (R♭)) 1-parameter group of Frobenius automs
+
The universal W -model W (H)u of a hyperfield H (when
exists) is the field of quotients of a natural ring:
H = hyperfield, {W (H)u, ρu, τu} = universal W -model
WZ(H) ⊂ W (H)u
integral subring generated by the τu(x)’s
(resp. WQ(H) ⊂ W (H)u
Then:
sub Q-algebra)
W (H)u ≃ Frac(WZ(H)) = Frac(WQ(H))
Example 1
S = {0, ±1} hyperfield of signs: WZ(S) ≃ Z, W (S) = Q
Example 2
♭)
WQ(R♭) = Q[R×
]
⊂
W
(
R
+
sub Q-algebra
!! NO universal W -model for K = {0, 1} Krasner hyperfield
IV. Universal pro-infinitesimal thickening of R
There is an archimedean counterpart of the ring homomorphism
of p-adic Hodge theory:
θ
1
b,+
B
:= W (OC♭ )[ p ] p
Cp,
∑
(0)
k
[xk ]p ) =
xk pk
θ(
k≫−∞
k≫−∞
∃! Q-algebra homomorphism
θ(
∑
i
ai[xi]) =
∑
ai(x0)i,
∑
♭ ) −→ R
Q[R×
]
=:
W
(
R
Q
+
θ([x]) = θ(τ (x)) = x0,
θ
∀x = (xn)n≥0 ∈ R♭
i
DEFINITION The universal pro-infinitesimal cover of R:
♭)/Ker(θ)n)
R∞ := lim
(W
(
R
Q
←−n
b,+ /(ker(θ))n )
(
B
is the archimedean counterpart of B+
dR := lim
←−n
THEOREM (1)
ℓ : R×
+→R
group homomorphism
Tℓ : WQ(R♭) −→ C ∞(R)
Tℓ(X)(z) :=
∑
aielog(xi)+(z−1)ℓ(xi)
i
Tℓ(X)(1)≡θ(X) =
(2)
∑
X :=
ring homomorphism
∑
ai[xi] ∈ WQ(R♭) = Q[R×
+]
i
i ai xi
θ
WQ(R♭) −→ R
The Taylor expansion at z = 1 induces the ring hom
T
ℓ
♭ )/Ker(θ)n −→
R∞ := lim
W
(
R
R[[z − 1]]
Q
←−n
(
)
d (T (1 − [x] − [1 − x])(z)
= −xℓ(x) − (1 − x)ℓ(1 − x)
dz ℓ
z=1
(3) {πp := [p] − p | p ∈ Z prime} ⊂ Ker(θ)/Ker(θ)2
linearly independent “periods” over R
THEOREM
(Ker(θ)/Ker(θ)2) is the infinite dimensional
R-linear space generated by the symbols
ε(x), x ∈ R, with relations:
(A)
ε(1 − x) = ε(x)
(B)
x ) + y ε(− x ) ,
ε(x + y) = ε(y) + (1 − y) ε( 1−y
y
(C)
x ε(1/x) = −ε(x) ,
∀y ∈
/ {0, 1}
∀x ̸= 0.
The only Lebesgue measurable group homomorphism
ℓ : R×
+→R
yields the linear form on Ker(θ)/Ker(θ)2:
s(x) := 1 − [x] − [1 − x] 7→ −x log(x) − (1 − x) log(1 − x) =: S(x)
given by the entropy function S(x)
V. Perfection of C
C♭ := {x = (xn)n≥0, convergent, xn ∈ C | (xn+1)3 = xn, ∀n}
(C
⨿
∼
n
C) ∪ {0} −→ C♭
ϵ e(z) 7→ (xn)n≥0 xn := ϵ ez/3 ; ϵ ∈ {±1}
(ϵ = limn→∞ xn, z = limn→∞ 3n(ϵxn − 1))
Product :
Sum :
ϵe(z) · ϵ′e(z ′) := ϵϵ′e(z + z ′)

ϵ e(z),




 ϵ′ e(z ′),
ϵ e(z) + ϵ′ e(z ′) :=

ϵ e(z),



 {0} ∪ {ϵ′′e(z ′′) | z ′′ ≤
C+ z},
C+ := {z ∈ C|ℜe(z) ≥ 0; ℑ(z) ≥ 0 if ℜ(z) = 0}
if
if
if
if
z ′ < C+ z
z < C+ z ′
z = z ′, ϵ = ϵ′
z = z ′, ϵ = −ϵ′
(total) cone
z ≤C+ z ′ ⇐⇒ z ′ − z ∈ C+
total order on C
C♭ is a perfect hyperfield of characteristic one
Embedding of hyperfields: R♭ ↩→ C♭
∀α ∈ (− π2 , π2 ], ∃ injective homom of hyperfields
Proposition
ι
α
R♭ −→
C♭ ,
(
B := {g =
)
φ
B −→ Aut(C♭),
φ
−→ Aut(C♭)
⊂
⊂
z∈C+
a 0
∈ GL2(R)|a, d > 0}
c d
FACT:
B
x 7→ sign(x)e(log |x| |{z}
eiα )
∼
♭)
R×
−→
Aut(
R
+
φ
(
isotropy group of C+
[φ(g)](ϵe(z)) = ϵe(g · z)
λ 0
gλ = 0 1
↑
λ
)
→
φgλ (ι0(x))
↑=
→ [φ(λ)](x) := sign(x)|x|λ
VI. The universal W -model of C♭
W (C♭) := {Frac(Q[C]), ρC+ , [·]}
THEOREM
ρC
+
Q[C] −→ C♭,
[·]
♭
C −→
ρC + (
Q[C],
universal W -model
n
∑
aj ϵj u(zj )) := ϵ0e(z0)
|{z}
j=1
∈Q×
+
[ϵ e(z)] := ϵ u(z)
z0 :=
∨
C+
(u(z) ∈ Q[C] canonical generator)
ρC+ ◦ [·] = id
ρC+ induces an isomorphism of hyperfields:
∼
W (C♭)/G −→ C♭,
G := Ker(ρC+ : W (C♭)× −→ (C♭)×)
∃! Q-algebra homomorphism
θ(
∑
i
ai[zi]) :=
∑
i
{zj }
θ
WQ(C♭) := Q[C] −→ C
aiev(zi) :=
∑
i
aiezi
The universal formal pro-infinitesimal thickening of C
is the Ker(θ)-adic completion of WQ(C♭):
♭ )/Ker(θ)n )
C∞ := lim
(W
(
C
Q
←−n
ℓ ∈ V := HomZ(C, C),
X=
∑
aiu(zi) ∈ WQ(C♭)
i
T : WQ(C♭) → F(V, C),
T (X)(ℓ) :=
∑
ℓ(zi ) ,
i ai e
θ(X) = T (X)(id)
THEOREM
The Taylor expansion at (id) ∈ V1 := HomR(C, C) ⊂ V
(−1,0)
of T (X)(L), for L = id + a(id) + b(id) ∈ V1 = V1
induces a surjective ring homomorphism:
ρ
C∞ −→ C[[a, b]],
a, b = variables
(0,−1)
⊕ V1
The “periods”: {πp := [e(log p)] − p ∈ Ker(θ)/Ker(θ)2, (p prime)}
are linearly independent over C.
There is another “period” which is purely complex:
it arises as the analogue of ε ∈ C♭p:
(ε(n) ∈ µpn (K̄))n≥0
generator of Zp(1)
ε := e(2πi) ∈ C♭
θ([ε]) = e2πi = 1,
ω := ([e(2πi)] − 1)/([e(πi)] − 1) ∈ Ker(θ)
ε1/2 = e(πi)
t := log([ε]) = −
∑ (1 − [ε])n
n≥1
n
ρ(t) = 2πi(a − b),
∈ C∞,
u(p) := log(
ρ(u(p)) = log(p)(a + b)
[p]
) ∈ C∞
p
R∞ θ
WZ(S)O = Z

/
Q = W (S )

/
C∞
θ

/
l
i i O
l
i
l
i i
l l
i i
l
i
l i
i i
l l
/F
tti i i
ul u l
/
1
/
R
?
F12 = {0, ±1}
ϵ
C
ρ
τ/F
C♭
ev

l
R♭
,
1
b b b b b1
b b b b b
b
b
b
b
b
b b b b b
b b b b b
b
b
b
b
b
b b b b b
S = {0, ±1}
∼
B+
dR
θ
W (Fp) = Zp l
/
Qp 
C♭p
m
mm
m
m
mm
vvm m
O
ˆ= C
(
Q̄
)
p
p
ρ
/
τ/F
1
,
?
Fp