The p -adic integral geometry formula Avi Kulkarni Dartmouth - PowerPoint PPT Presentation

The p -adic integral geometry formula Avi Kulkarni Dartmouth College August 27, 2020 Joint work with Antonio Lerario

Part I: The real world

Integral geometry deals with averaging metric properties under the action of a Lie group. Applications in: representation theory, convex geometry, random algebraic geometry, etc.

The metric Let S n → P n be the Hopf fibration. We define d ( x , y ) = | sin( angle between the points ) | this descends to a metric on P n . The volume form on the sphere restricts to projective space.

The volume If Y is a codimension m − k submanifold of a Riemannian manifold M whose volume density is vol and � U ( Y , ǫ ) = B ( x , ǫ ) x ∈ Y denotes the ǫ -neighborhood of Y in M , then vol ( U ( Y , ǫ )) vol k ( Y ) := lim vol ( B R m − k ( 0 , ǫ )) . ǫ → 0

Theorem Let X , Y ⊆ P n ( R ) be real algebraic sets. Then vol z ( X ∩ gY ) dg = vol x X vol x P x · vol y Y � vol z P z vol y P y SO n + 1 ( R ) where x = dim X , y = dim Y , z = dim( X ∩ gY ) . Corollary The expected number of zeros of a random real polynomial of √ degree d is d, with respect to a certain distribution.

Theorem Let X , Y ⊆ P n ( R ) be real algebraic sets. Then vol z ( X ∩ gY ) dg = vol x X vol x P x · vol y Y � vol z P z vol y P y SO n + 1 ( R ) where x = dim X , y = dim Y , z = dim( X ∩ gY ) . Corollary The expected number of zeros of a random real polynomial of √ degree d is d, with respect to a certain distribution.

Act II: Leaving the real world.

Definition The p -adic absolute value is defined on Q \{ 0 } by � p e a � � p := p − e e ∈ Z and gcd( p , ab ) = 1 , | 0 | p := 0 . � � b �    a j p j : e ∈ Z , a e � = 0 , a j ∈ { 0 , . . . , p − 1 }  � Q p =   j ≥ e with the ring structure given by “Laurent series arithmetic”.

◮ | a + b | p ≤ max {| a | p , | b | p } , with equality when this maximum is attained exactly once. ( Ultrametric inequality ). ◮ | n | p ≤ 1 for all n ∈ Z . ◮ |·| p : Q → R has image { 0 } ∪ { p − n : n ∈ Z } . ◮ ( Q p , |·| p ) is a metric space. ◮ For r ≥ 0, define B ( x ; r ) = { y ∈ Q p : | y − x | p ≤ r } . ◮ Q p is a locally compact topological space, and ring operations are continuous. Definition The ring of p-adic integers is defined as Z p := { x ∈ Q p : | x | p ≤ 1 } .

The additive group ( Q p , +) is a locally compact topological group. Thus there is a Haar measure. We normalize µ ( Z p ) = 1. Lemma B ( x ; r ) is closed and open in the metric topology. (When r > 0 ) Lemma Let y ∈ B ( x ; r ) . Then B ( x ; r ) = B ( y ; r ) . Corollary Z p is the disjoint union of the open and closed balls B ( a ; p − 1 ) , a ∈ { 0 , 1 , . . . , p − 1 } . Therefore µ ( B ( a ; p − 1 )) = p − 1 , and µ ( Z × p ) = 1 − p − 1 . ( > 0 !!)

Critical detail: Q p has a totally disconnected topology!!!

Define the padic unit sphere by Q p := { x ∈ Q pn + 1 : � x � p = 1 } . S n Remark The dimension of S n is n + 1. Here is a picture: ( − 1 , 1 ) ( 0 , 1 ) ( 1 , 1 ) ( − 1 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( − 1 , − 1 ) ( 0 , − 1 ) ( 1 , − 1 )

Q p := { x ∈ Q pn + 1 : � x � p = 1 } . S n We have the Hopf fibration ϕ : ( a 0 , . . . , a n ) �→ ( a 0 : . . . : a n ) The spherical metric d ( x , y ) := � x ∧ y � p gives P n ( Q p ) the structure of a metric space. ( R : the sine of the angle between x , y ∈ S n R is ±� x ∧ y � R .)

Definition µ ( U ) := µ ( Z p × ) − 1 µ ( ϕ − 1 ( U )) . Proposition A maximal compact subgroup of GL n + 1 ( Q p ) is GL n + 1 ( Z p ) . It is unique up to conjugation. The metric on the unit sphere is GL n + 1 ( Z p ) -invariant.

Definition Let X ⊆ P n . For each m define (mod p m ) : x ∈ ϕ − 1 ( X ) } N m ( X ) := # { x . p m ( 1 − p − 1 ) The d -dimensional volume of X ⊆ P n Q p is N m ( X ) vol d ( X ) := lim . p md m →∞ This definition of volume comes from the theory of zeta functions studied by Denef, Igusa, Oesterlé, Serre, etc.

Definition Let X ⊆ P n . For each m define (mod p m ) : x ∈ ϕ − 1 ( X ) } N m ( X ) := # { x . p m ( 1 − p − 1 ) The d -dimensional volume of X ⊆ P n Q p is N m ( X ) vol d ( X ) := lim . p md m →∞ Proposition (Serre) If X is d-dimensional, then the limit in the definition exists and is finite.

Definition Let X ⊆ P n . For each m define (mod p m ) : x ∈ ϕ − 1 ( X ) } N m ( X ) := # { x . p m ( 1 − p − 1 ) The d -dimensional volume of X ⊆ P n Q p is N m ( X ) vol d ( X ) := lim . p md m →∞ Lemma This is also the p-adic tube volume. i.e, � � � m →∞ p m ( n − a ) · µ n B ( x , p − m ) vol a ( X ) = lim . x ∈ X

Lemma This is also the p-adic tube volume. i.e, � � � m →∞ p m ( n − a ) · µ n B ( x , p − m ) vol a ( X ) = lim . x ∈ X Proof. Consider the continuous map (mod p m ) : x ∈ X } . π m : X → { x Choosing an arbitrary set of lifts, we obtain a list of centers needed to cover X .

Theorem Let X be a subscheme of P n which is smooth over Spec Z p . Then the d-dimensional volume is also the Weil canonical volume of X = X ( Z p ) . Proof. By smoothness, the Jacobian is non-vanishing modulo p at every point. We then use a quantitative inverse function theorem to count points modulo p m . N m ( X ) = X ( F p ) lim p md p d m →∞ This turns out to be equal to the Weil canonical volume.

Examples in P 2 ( Q 2 ) : � � 1 + 1 vol 1 Z ( x 2 + y 2 − z 2 ) = 1 , vol 1 Z ( x − y + z ) = , 2 � 1 + 1 � vol 1 Z ( 2 ( y 2 − xz )) = . 2

Act III: p -adic integral geometry

Theorem (K.-Lerario) Let X , Y ⊆ P n ( Q p ) be algebraic sets. Then � vol z ( X ∩ gY ) dg = vol x X vol x P x · vol y Y vol z P z vol y P y GL n + 1 ( Z p ) where x = dim X , y = dim Y , z = dim( X ∩ gY ) .

Proof sketch Lemma (Hensel’s lemma) Let f = ( f 1 , . . . , f n ) ∈ Z p [ x 1 , . . . , x n ] n , let a ∈ Z pn , and let J f ( a ) be the Jacobian matrix of f at a. If � f ( a ) � < | J f ( a ) | 2 , then there is a unique α ∈ Z pn such that (mod p m ) , f ( α ) = 0 and α ≡ a where m := 1 − log p | J f ( a ) | .

Lemma (Linear Approximation Lemma) Let X 1 , . . . , X s ⊆ P n be algebraic sets such that j = 1 codim ( X j ) = n . Let x ( j ) ∈ X j be smooth points, and denote � s by U x ( j ) balls of P n of radius p − m centered at these points. Assume that in these balls we have local equations U x ( j ) ∩ X j = { f x ( j ) = 0 } . � 2 > p − m , then � f 1 ( x ( 1 ) ) , . . . , f s ( x ( s ) ) �� � If � J s s � � # ( X j ∩ U x ( j ) ) = # ( T x ( j ) X j ∩ U x ( j ) ) = 0 or 1 . j = 1 j = 1 Proof. Apply Hensel’s lemma in each ball.

Proposition For A an open compact subset of an a-dimensional algebraic set in P n Q p , we have � vol k ( A ∩ gH ) vol a ( A ) dg = Q p ) . vol a ( P a vol k ( P k Q p ) GL n + 1 ( Z p )

� # ( A ∩ gL ) dg GL n + 1 ( Z p ) Convert a variety into a union of many tangent spaces (Linear approximation lemma) N m ( A ) � � B ( u i , p − m ) ∩ T u i A i ∩ gL � � = lim # m →∞ GL n + 1 ( Z p ) i = 1 Prove the result for pieces of linear varieties, then ℓ →∞ N m ( A ) · vol a ( B ( u , p − m ) ∩ P a ) = lim vol a ( P a ) Use the definition of volume vol a ( A ) = Q p ) . vol a ( P a

Applications Theorem (Oesterlé, weak form) For an equidimensional variety A ⊆ P n Q p of dimension d, we have Q p ) · p md + o ( 1 ) . N m ( A ) ≤ deg( A ) vol d ( P d Proof. vol k ( A ∩ gH ) vol a ( A ) � dg = Q p ) . vol a ( P a vol k ( P k Q p ) GL n + 1 ( Z p ) The integrand is at most the degree almost everywhere.

Corollary Let g ( t ) := ζ 0 + ζ 1 t + . . . + ζ d t d . The expected number of zeros of g ( t ) in Q p is 1 . Proof. Check that the standard Veronese is an isometry. Applying the Integral geometry formula gives the result.

Theorem (Evans, univariate) Let � t � � t � g ( t ) := ζ 0 + ζ 1 + . . . + ζ d , 1 d where { ζ k } d k = 0 is a family of i.i.d. uniform variables in Z p . Then the expected number of zeroes of g contained in Z p is 1 + p − 1 � − 1 p ⌊ log p d ⌋ � . | d | p p − 1 The expected number of zeros outside the unit disk is 1 + p − 1 .

Proof. Let � � t � � t �� F : ( t , 1 ) → 1 , , . . . , 1 d be the Mahler Veronese map. The Jacobian of F is � 0 , 1 , . . . , d � t �� . dt d For t ∈ Z p the absolute value of the largest entry is exactly p ⌊ log p d ⌋ . The Hopf fibration restricted to the image of F is an isometry, so we can apply the integral geometry formula.

Thanks! p -adic Integral Geometry, arXiv:1908.04775