A p -adically entire function with integral values on Q p and - PowerPoint PPT Presentation

A p -adically entire function with integral values on Q p and additive characters of perfectoid fields Francesco Baldassarri (Padova) Papeete, August 24, 2015

Outline A p -adic entire function Integrality of Ψ p Additive characters of perfectoid fields Barsotti-Witt constructions Hyperexponential vectors Universal topological Hopf algebras

The function Ψ p A prime p is fixed all over. We consider the formal solution ∞ � a i T i ∈ Z [[ T ]] , Ψ( T ) = Ψ p ( T ) = T + i = 2 to the functional equation ∞ � p − j Ψ( p j T ) p j = T . ( ∗ ) j = 0 The following facts were proven in my thesis (Padova 1974 - Ann. Sc. Norm. Sup. 1975)

1. Ψ p is p -adically entire; 2. Ψ p ( Q p ) ⊂ Z p ; 3. for any i ∈ Z and x ∈ Q p , if we define x − i := Ψ p ( p i x ) mod p ∈ F p then ∞ � [ x i ] p i ∈ W ( F p )[ 1 / p ] = Q p , x = i >> −∞ where [ t ] , for t ∈ F p , is the Teichm¨ uller representative of t in W ( F p ) = Z p . 4. Ψ p trivializes the addition law of Witt covectors with coefficients in the Fr´ echet algebra Q p { x , y } of entire functions of x and y .

( . . . , Ψ( p 2 x ) , Ψ( px ) , Ψ( x )) + ( . . . , Ψ( p 2 y ) , Ψ( py ) , Ψ( y )) = ( . . . , Ψ( p 2 ( x + y )) , Ψ( p ( x + y )) , Ψ( x + y )) . Watch out : This is a sum of Witt covectors ! We will explain later what this means. Some of these results admit an elementary proof. For example Proposition The functional equation ( ∗ ) has a unique solution Ψ = Ψ p ∈ T Z [[ T ]] .

Proof. We endow T Z [[ T ]] of the T -adic topology. It is clear that, for any ϕ ∈ T Z [[ T ]] , the series T − � ∞ j = 1 p − j ϕ ( p j T ) p j converges in T Z [[ T ]] and that the map ∞ � p − j ϕ ( p j T ) p j , ϕ �− → T − j = 1 is a contraction of the complete metric space T Z [[ T ]] . So, this map has a unique fixed point which is Ψ( T ) . It is also easy to prove that Proposition The series Ψ( T ) is entire.

Proof. Since Ψ ∈ T Z [[ T ]] , we deduce that Ψ converges for v p ( T ) > 0. On the other hand, it is clear that the coefficient of T in Ψ( T ) is 1. Therefore, for v p ( T ) > 0, v p (Ψ( T )) = v p ( T ) . Suppose Ψ converges for v p ( T ) > ρ . Then, for j ≥ 1, Ψ( p j T ) p j converges for v p ( T ) > ρ − 1. Moreover, if j > − ρ + 1 and v p ( T ) > ρ − 1, we have v p ( p − j Ψ( p j T ) p j ) ≥ − j + p j ( j + ρ − 1 ) , and this last term → + ∞ , as j → + ∞ . This shows that the series T − � ∞ j = 1 p − j Ψ( p j T ) p j converges uniformly for v p ( T ) > ρ − 1, so that its sum, which is Ψ , is analytic for v p ( T ) > ρ − 1. It follows immediately from this that Ψ is an entire function.

Outline A p -adic entire function Integrality of Ψ p Additive characters of perfectoid fields Barsotti-Witt constructions Hyperexponential vectors Universal topological Hopf algebras

Proposition For any a ∈ Q p , Ψ p ( a ) ∈ Z p . Proof. Let a ∈ Z p . We define by induction the sequence { a i } i = 0 , 1 ,... : 0 − a p 2 a 0 = a , a 1 = p − 1 ( a 0 − a p 0 ) , a 2 = p − 2 ( a p 0 ) + p − 1 ( a 1 − a p 1 ) , i − 1 � p j − i ( a p i − j − 1 − a p i − j a i = ) . j j j = 0 Since, for any a , b ∈ Z p , if a ≡ b mod p , then a p n ≡ b p n mod p n + 1 , while a ≡ a p mod p , we see that a i ∈ Z p , for any i .

We then see by induction that, for any i , i − 1 i � � p j a p i − j p j a p i − j a i = p − i ( a − ) or, equivalently, a = . j j j = 0 j = 0 More precisely, if we stick in the formula which defines a i , namely i − 1 i − 1 � � p j a p i − j − 1 p j a p i − j p i a i = − j j j = 0 j = 0 i − 1 � p j a p i − j − 1 the ( i − 1 ) -st step of the induction, namely, a = , we j j = 0 get i − 1 � p j a p i − j p i a i = a − , j j = 0 which is precisely the i -th inductive step.

From the functional equation we have i � p − ℓ Ψ( p ℓ p − i a ) p ℓ = Ψ( p − i a ) ≡ p − i a − ℓ = 1 i − 1 � p j Ψ( p − j a ) p i − j ) p − i ( a − mod p Z p . j = 0 We then see by induction that Ψ( p − i a ) ≡ a i mod p Z p , which proves the statement. In fact, assume Ψ( p − j a ) ≡ a j mod p Z p , for j = 0 , 1 , . . . , i − 1, and plug this information in the previous formula. We get i i − 1 � � p j a p i − j p − ℓ a p ℓ Ψ( p − i a ) ≡ p − i a − i − ℓ = p − i ( a − ) = a i , j ℓ = 1 j = 0 which is the i -th inductive step.

We now know more. Theorem The valuation polygon of Ψ p µ �− → v ( f , µ ) = inf i ∈ Z i µ + v ( a i ) goes through the origin, has slope 1 for µ > − 1 , and slope p j , for − j − 1 < µ < − j, j = 1 , 2 , . . . .

✻ � � � slope 1 − 3 − 2 − 1 µ -line � ✲ • • • • � � ✁ ✁ slope p ✁ ✁ ✄ ✄ ✄ ✄ slope p 2 ✄ ✄ ✄ ✄ Figure : The valuation polygon of Ψ p .

Corollary The Newton polygon Nw (Ψ) has vertices at the points V i := ( − p i , i p i − p i − 1 p − 1 ) = ( − p i , i p i − p i − 1 − · · · − p − 1 ) . The equation of the side joining the vertices V i and V i − 1 is Y = − iX − p i − 1 p − 1 ; its projection on the X-axis is the segment [ − p i , − p i − 1 ] . So, Nw (Ψ) has the form described in the next figure.

∗ ❆ vertex V 2 at ( − p 2 , 2 p 2 − p − 1 ) ❆ ✻ ❆ ❆ ❆ slope -2 ❆ vertex V i at ( − p i , i p i − p i − 1 − · · · − p − 1 ) ❆ ❆ ❆ ❆ ❆ ❆ ∗ ❅ vertex V 1 at ( − p , p − 1 ) ❅ slope -1 ❅ − p 2 − p ❅ − 1 ❅ ✲ • • • Figure : The Newton polygon Nw (Ψ p ) of Ψ p .

Corollary For any i = 0 , 1 , . . . , the map Ψ = Ψ p induces finite coverings of degree p i , → { x ∈ C p | v p ( x ) > − p i + 1 − 1 Ψ : { x ∈ C p | v p ( x ) > − i − 1 } − } , p − 1 (in particular, an isomorphism ∼ Ψ : { x ∈ C p | v p ( x ) > − 1 } − − → { x ∈ C p | v p ( x ) > − 1 } , for i = 0 ).

More precisely, Ψ induces finite maps of degree p i Ψ : { x ∈ C p | − ( i + 1 ) < v p ( x ) < − i } − → { x ∈ C p | − p i + 1 − 1 < v p ( x ) < − p i − 1 p − 1 } , p − 1 and finite maps of degree p i + 1 − p i → { x ∈ C p | − p i + 1 − 1 Ψ : { x ∈ C p | v p ( x ) = − i − 1 } − ≤ v p ( x ) } . p − 1

The function Ψ p : A 1 Q p → A 1 Q p is a quasi-finite covering of the Berkovich affine line over Q p by itself. Aside from ramification, its behaviour is very similar to the one of the map log : D Q p ( 1 , 1 − ) → A 1 Q p , where D Q p ( 1 , 1 − ) is the open unit disk in A 1 Q p . I believe, but cannot prove, that, after base change to C p , Ψ p is a (ramified) Galois abelian covering.

Outline A p -adic entire function Integrality of Ψ p Additive characters of perfectoid fields Barsotti-Witt constructions Hyperexponential vectors Universal topological Hopf algebras

Perfectoid fields We recall that a perfectoid field is a non-discretely valued non-archimedean field K such that the Frobenius map of K ◦ / pK ◦ is surjective. For any perfectoid field ( K , | | ) , one defines the tilt K ♭ = lim ← ( K , x �→ x p ) of K . It is a perfect � non-archimedean extension of � K , K ♭ = � K (( t 1 / p ∞ )) . The t -adic valuation of the element ̟ = ( ̟ ( 0 ) , ̟ ( 1 ) , . . . ) ∈ K ♭ , with ̟ ( i ) ∈ K , ( ̟ ( i + 1 ) ) p = ̟ ( i ) , is, by definition, v K ( ̟ ( 0 ) ) . If K is of characteristic p , then K ♭ = K .

A pseudo-uniformizer ̟ = ( ̟ ( 0 ) ← ̟ ( 1 ) ← . . . ) of K ♭ is an element of ( K ♭ ) ◦◦ . For any i = 0 , 1 , 2 , . . . , we define ̟ i = ( ̟ ( i ) ← ̟ ( i + 1 ) ← . . . ) , so that ̟ i is the unique p i -th root of ̟ in K ♭ . We consider the element � � ̟ ( i ) p i + ( ̟ ( 0 ) ) p − i p i ∈ K . π = π ( ̟ ) := i ≥ 0 i < 0 Notice that this is a convergent sum in K and that π ( ̟ p i ) = p i π ( ̟ ) , π ( ̟ i ) = p − i π ( ̟ ) .

We will use the formula of Dieudonn´ e ∞ ∞ ∞ � � � x ( i ) T p i = 1 + g i ( x 0 , x 1 , . . . , x [ log p i ] ) T i , F ( x i T p i ) = exp i = 0 i = 0 i = 1 where F ( T ) = exp ( � ∞ i = 0 T p i / p i ) ∈ Z ( p ) [[ T ]] is the Artin-Hasse exponential series and x ( i ) = � i n = 0 p n − i x p n − i is the ghost n component of the Witt vector ( x 0 , x 1 , . . . ) . The plan is to introduce variables x i , y i with negative indices i and to prolong that formula into (for S = Z [ 1 / p ] ∩ R ≥ 0 ) ∞ ∞ � � x ( i ) T p i = F ( x i T p i ) = exp i = −∞ i = −∞ � g q ( . . . , x [ log q ] − 1 , x [ log q ] ) T q . 1 + q ∈ S

Then, we want to specialize x i �→ Ψ p ( p − i x ) , for any i ∈ Z , and T 1 / p i �→ ̟ ( i ) , for any i = 0 , 1 , 2 , . . . and to use the integrality properties of Ψ p (i.e. Ψ p ( Q p ) ⊂ Z p ), to show that the map x �→ exp π ( ̟ ) x , a priori only defined for 1 v p ( x ) > p − 1 − v p ( π ( ̟ )) , canonically extends to a continuous additive character Ψ ̟ : Q p → 1 + K ◦◦ . Note that such a character is an element of the inverse limit of 1 + K ◦◦ under the p -th power map, which is the same as 1 + ( K ♭ ) ◦◦ . We then obtain a map ̟ �→ Ψ ̟ from the open unit disk at 0 to the open disk at 1, both over K ♭ .