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 Sn → Pn be the Hopf fibration. We define d(x, y) = |sin(angle between the points)| this descends to a metric on Pn. 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) =

• x∈Y

B(x, ǫ) denotes the ǫ-neighborhood of Y in M, then volk(Y) := lim

ǫ→0

vol(U(Y, ǫ)) vol(BRm−k(0, ǫ)).

Theorem

Let X, Y ⊆ Pn(R) be real algebraic sets. Then

• SOn+1(R)

volz(X ∩ gY) volzPz dg = volxX volxPx · volyY volyPy 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 ⊆ Pn(R) be real algebraic sets. Then

• SOn+1(R)

volz(X ∩ gY) volzPz dg = volxX volxPx · volyY volyPy 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

• pe a

b

• p := p−e

e ∈ Z and gcd(p, ab) = 1, |0|p := 0. Qp =   

• j≥e

ajpj : e ∈ Z, ae = 0, aj ∈ {0, . . . , p − 1}    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}. ◮ (Qp, |·|p) is a metric space. ◮ For r ≥ 0, define B(x; r) = {y ∈ Qp : |y − x|p ≤ r}. ◮ Qp is a locally compact topological space, and ring

• perations are continuous.

Definition

The ring of p-adic integers is defined as Zp := {x ∈ Qp : |x|p ≤ 1}.

The additive group (Qp, +) is a locally compact topological

• group. Thus there is a Haar measure. We normalize µ(Zp) = 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

Zp 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: Qp has a totally disconnected topology!!!

Define the padic unit sphere by Sn

Qp := {x ∈ Qpn+1 : xp = 1}.

Remark

The dimension of Sn is n + 1. Here is a picture:

(−1, −1) (−1, 0) (−1, 1) (0, −1) (0, 0) (0, 1) (1, −1) (1, 0) (1, 1) (0, 0)

Sn

Qp := {x ∈ Qpn+1 : xp = 1}.

We have the Hopf fibration ϕ: (a0, . . . , an) → (a0 : . . . : an) The spherical metric d(x, y) := x ∧ yp gives Pn(Qp) the structure of a metric space. (R: the sine of the angle between x, y ∈ Sn

R is ±x ∧ yR.)

Definition

µ(U) := µ(Zp×)−1µ(ϕ−1(U)).

Proposition

A maximal compact subgroup of GLn+1(Qp) is GLn+1(Zp). It is unique up to conjugation. The metric on the unit sphere is GLn+1(Zp)-invariant.

Definition

Let X ⊆ Pn. For each m define Nm(X) := #{x (mod pm) : x ∈ ϕ−1(X)} pm(1 − p−1) . The d-dimensional volume of X ⊆ Pn

Qp is

vold(X) := lim

m→∞

Nm(X) pmd . This definition of volume comes from the theory of zeta functions studied by Denef, Igusa, Oesterlé, Serre, etc.

Definition

Let X ⊆ Pn. For each m define Nm(X) := #{x (mod pm) : x ∈ ϕ−1(X)} pm(1 − p−1) . The d-dimensional volume of X ⊆ Pn

Qp is

vold(X) := lim

m→∞

Nm(X) pmd .

Proposition (Serre)

If X is d-dimensional, then the limit in the definition exists and is finite.

Definition

Let X ⊆ Pn. For each m define Nm(X) := #{x (mod pm) : x ∈ ϕ−1(X)} pm(1 − p−1) . The d-dimensional volume of X ⊆ Pn

Qp is

vold(X) := lim

m→∞

Nm(X) pmd .

Lemma

This is also the p-adic tube volume. i.e, vola(X) = lim

m→∞ pm(n−a) · µn x∈X

B(x, p−m)

• .

Lemma

This is also the p-adic tube volume. i.e, vola(X) = lim

m→∞ pm(n−a) · µn x∈X

B(x, p−m)

• .

Proof.

Consider the continuous map πm : X → {x (mod pm) : 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 Pn which is smooth over Spec Zp. Then the d-dimensional volume is also the Weil canonical volume of X = X(Zp).

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 pm. lim

m→∞

Nm(X) pmd = X(Fp) pd This turns out to be equal to the Weil canonical volume.

Examples in P2(Q2): vol1Z(x − y + z) =

• 1 + 1

2

• ,

vol1Z(x2 + y2 − z2) = 1, vol1Z(2(y2 − xz)) =

• 1 + 1

2

• .

Act III: p-adic integral geometry

Theorem (K.-Lerario)

Let X, Y ⊆ Pn(Qp) be algebraic sets. Then

• GLn+1(Zp)

volz(X ∩ gY) volzPz dg = volxX volxPx · volyY volyPy where x = dim X, y = dim Y, z = dim(X ∩ gY).

Proof sketch

Lemma (Hensel’s lemma)

Let f = (f1, . . . , fn) ∈ Zp[x1, . . . , xn]n, let a ∈ Zpn, and let Jf(a) be the Jacobian matrix of f at a. If f(a) < |Jf(a)|2 , then there is a unique α ∈ Zpn such that f(α) = 0 and α ≡ a (mod pm), where m := 1 − logp |Jf(a)|.

Lemma (Linear Approximation Lemma)

Let X1, . . . , Xs ⊆ Pn be algebraic sets such that s

j=1 codim(Xj) = n. Let x(j) ∈ Xj be smooth points, and denote

by Ux(j) balls of Pn of radius p−m centered at these points. Assume that in these balls we have local equations Ux(j) ∩ Xj = {fx(j) = 0}. If

• J
• f1(x(1)), . . . , fs(x(s))
• 2 > p−m, then

#

s

• j=1

(Xj ∩ Ux(j)) = #

s

• j=1

(Tx(j)Xj ∩ Ux(j)) = 0 or 1.

Proof.

Apply Hensel’s lemma in each ball.

Proposition

For A an open compact subset of an a-dimensional algebraic set in Pn

Qp, we have

• GLn+1(Zp)

volk(A ∩ gH) volk(Pk

Qp)

dg = vola(A) vola(Pa

Qp).

• GLn+1(Zp)

# (A ∩ gL) dg Convert a variety into a union of many tangent spaces (Linear approximation lemma) = lim

m→∞

• GLn+1(Zp)

Nm(A)

• i=1

#

• B(ui, p−m) ∩ TuiAi ∩ gL
• Prove the result for pieces of linear varieties, then

= lim

ℓ→∞ Nm(A) · vola(B(u, p−m) ∩ Pa)

vola(Pa) Use the definition of volume = vola(A) vola(Pa

Qp).

Applications

Theorem (Oesterlé, weak form)

For an equidimensional variety A ⊆ Pn

Qp of dimension d, we

have Nm(A) ≤ deg(A)vold(Pd

Qp) · pmd + o(1).

Proof.

• GLn+1(Zp)

volk(A ∩ gH) volk(Pk

Qp)

dg = vola(A) vola(Pa

Qp).

The integrand is at most the degree almost everywhere.

Corollary

Let g(t) := ζ0 + ζ1t + . . . + ζdtd. The expected number of zeros of g(t) in Qp is 1.

Proof.

Check that the standard Veronese is an isometry. Applying the Integral geometry formula gives the result.

Theorem (Evans, univariate)

Let g(t) := ζ0 + ζ1 t 1

• + . . . + ζd

t d

• ,

where {ζk}d

k=0 is a family of i.i.d. uniform variables in Zp. Then

the expected number of zeroes of g contained in Zp is p⌊logp d⌋ 1 + p−1−1 . The expected number of zeros outside the unit disk is

|d|pp−1 1+p−1 .

Proof.

Let F : (t, 1) →

• 1,

t 1

• , . . . ,

t d

• be the Mahler Veronese map. The Jacobian of F is
• 0, 1, . . . , d

dt t d

• .

For t ∈ Zp the absolute value of the largest entry is exactly p⌊logp 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