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Geometry of p-adic

Zair Ibragimov CSUF

Valuations on Geometry of p-adic numbers Rational Numbers

Completions of Rational Numbers Zair Ibragimov

The p-adic CSUF numbers

Geometry of p-adic Talk at Fullerton College July 14, 2011 Valuations on Rational Numbers

Geometry of p-adic Let = {..., −2, −1, 0, 1, 2,... } denote the integers. Let numbers Z = {a/b : a, b ∈ and b > 0} denote the rational numbers. Zair Ibragimov Q Z CSUF

Valuations on We can add and multiply rational numbers: Rational Numbers a/b + c/d = (ad + bc)/(bd) and (a/b) · (c/d) = (ac)/(bd)

Completions of Rational Numbers By a on Q we mean a function v : Q → [0, +∞) The p-adic satisfying the following three properties: numbers

Geometry of 1 v(x) ≥ 0, and v(x) = 0 if and only if x = 0; p-adic integers 2 v(x + y) ≤ v(x) + v(y) (the triangle inequality); 3 v(xy) = v(x)v(y).

We say that a valuation v is non-Archimedean if it satisfies the strong triangle inequality: v(x + y) ≤ max{v(x), v(y)}. The standard valuation on Q

Geometry of p-adic The real function | · | on is a well-known numbers Q example of a valuation. Zair Ibragimov CSUF Recall that |x| = x if x ≥ 0 and |x| = −x if x < 0. Valuations on Rational Properties (1) and (3) are easily verified. Numbers

Completions To show property (2), enough to consider three cases: of Rational Numbers Case 1: x, y ≥ 0. Then |x + y| = x + y = |x| + |y|. The p-adic numbers Case 2: x, y ≤ 0. Then |x + y| = −x − y = |x| + |y|. Geometry of p-adic integers Case 3: x ≤ 0 and y ≥ 0. Then |x + y| is either x + y or −x − y and |x| + |y| = −x + y. Hence |x + y| ≤ |x| + |y|.

The function | · |0 (defined by |x|0 = 0 if x = 0 and |x|0 = 1 if x 6= 0) is easily seen to be a valuation on Q. It is referred to as the trivial valuation on Q. Fundamental Theorem of Arithmetics

Geometry of p-adic numbers Theorem (Fundamental Theorem of Arithmetics) Zair Ibragimov CSUF Any can be written as a product of primes. That is,

Valuations on a a a Rational |n| = p 1 · p 2 ··· p k , Numbers 1 2 k

Completions of Rational where p1, p2,..., pk are primes and a1, a2,..., ak are positive Numbers integers. The p-adic numbers For example, 360 = 22 · 32 · 5 and 93555 = 34 · 52 · 7 · 11. Geometry of p-adic integers

If we allow a1, a2,..., ak to be any integers (positive or negative), then we can write any rational as a product a1 a2 ak of primes. That is, |x| = p1 · p2 ··· pk . 360 2 −2 −1 −1 −1 For example, 93555 = 2 · 3 · 5 · 7 · 11 . The p-adic valuation on Q

Geometry of Let p be a fixed , i.e., p = 2, 3, 5, 7, 11, 13,... . p-adic numbers Any non-zero rational number x can be written uniquely as Zair Ibragimov a CSUF x = pn , b Valuations on Rational where neither a not b is divisible by p and n ∈ Z. The p-adic Numbers absolute value on is defined by Completions Q of Rational ( Numbers 0 if x = 0 |x|p = The p-adic −n numbers p if x 6= 0.

Geometry of 2 −2 −1 −1 −1 p-adic integers For example, since 360/93555 = 2 · 3 · 5 · 7 · 11 , −2 2 |360/93555|2 = 2 , |360/93555|3 = 3 , |360/93555|7 = 7 and 0 |360/93555|p = p = 1 for all p 6= 2, 3, 5, 7, 11. Problem Show that the p-adic absolute value is a non-Archimedean valuation on Q. Show that |x|p ≤ 1 if and only if x ∈ Z. Ostrowski’s Theorem

Geometry of p-adic Two valuations v1 and v2 on Q is said to be equivalent if there numbers exists c > 0 such that Zair Ibragimov CSUF c v2(x) = v1(x) for all x ∈ . Valuations on Q Rational Numbers Completions Theorem (Ostrowski) of Rational Numbers Any valuation on Q is equivalent to either the trivial valuation The p-adic numbers or the real absolute value | · | or a p-adic absolute value | · |p for Geometry of some prime p. p-adic integers

The classical Analysis, and Geometry is based on the standard valuation on Q. The p-adic valuation, which is the topic of my talk, has been introduced in 1897 by Kurt Hensel. A systematic study of p-adic numbers began in the late 1960’s. Metrics and Ultrametrics on Q

Geometry of p-adic A distance function d : Q × Q → [0, +∞) is called a metric on numbers if (1) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y, Zair Ibragimov Q CSUF (2) d(x, y) = d(y, x) and (3) d(x, y) ≤ d(x, z) + d(z, y).

Valuations on It is called an ultrametric if it satisfies (1), (2) and Rational Numbers (3’): d(x, y) ≤ max{d(x, z), d(z, y)}.

Completions of Rational Each valuation v on Q induces a metric d to measure a Numbers distance between points in Q by d(x, y) = v(x − y). The p-adic numbers The real absolute value | · | induces Euclidean metric on Q, Geometry of p-adic integers given by d(x, y) = |x − y|. As we mentioned above, much of classical analysis and geometry is based on Euclidean metric. Problem

Show that the distance function d(x, y) = |x − y|p on Q induced by the p-adic absolute value | · |p is an ultrametric. Completions of Q

Geometry of The rational numbers , equipped with Euclidean metric or p-adic Q numbers p-adic ultrametric, is not adequate to study problems in Zair Ibragimov CSUF Algebra or Analysis. 2 Valuations on For example, the x − 2 = 0 in Q is not Rational √ Numbers solvable in Q because 2 is not a rational number. Completions of Rational Similarly, the Numbers ∞ 1 The p-adic X numbers k! k=0 Geometry of p-adic integers in Q does not converge in Q because its sum is equal to e, which is not a rational number.

Overcoming these problems require a completion of Q. In other words, we need to add more points to Q. Completions of Q depend on the metric used and is based on the notion of Cauchy . Completion of Q with respect to Euclidean metric

Geometry of We say that a {x } of numbers in is a Cauchy p-adic n Q numbers sequence if |xn − xm| → 0 as n, m → ∞. For example, a trivial Zair Ibragimov CSUF sequence x, x,... is a . So is the sequence 1, 1/2, 1/3,... . Valuations on Rational Numbers Two Cauchy sequences {xn} and {yn} are said to be equivalent Completions if |xn − yn| → 0 as n → ∞. Let [{xn}] denote the of all of Rational Numbers Cauchy sequences in Q that are equivalent to {xn}. The p-adic numbers The set of all equivalent Cauchy sequences, equipped with the Geometry of metric d([{x }], [{y }]) = lim |x − y |, is called the p-adic integers n n n→∞ n n metric completion of Q. This space can be identified with the real numbers R. Problem √ Show that a sequence converging to 2 is a Cauchy sequence. Show that all such sequences are equivalent. Completion of Q with respect to p-adic ultrametric

Geometry of p-adic The p-adic numbers, denoted by p, is obtained as the numbers Q completion of the rational numbers with respect to the Zair Ibragimov Q CSUF p-adic ultrametric.

Valuations on Rational Since every rational numbers x can be identified with a Cauchy Numbers sequence {x, x,... }, every rational number is a p-adic number. Completions of Rational That is, Q ⊂ Qp. But there are much more p-adic numbers Numbers that cannot be identified with rational numbers. In fact, while The p-adic numbers the rational numbers Q are countable, the p-adic numbers Qp Geometry of are uncountable. p-adic integers There are significant differences between the two completions of Q due to the fact that Euclidean metric satisfies only the triangle inequality while the p-adic metric satisfies the strong triangle inequality. For example, R is a while Qp is totally disconnected like the middle-third . Completion of Z with respect to p-adic ultrametric

Geometry of p-adic numbers The integers Z, equipped with Euclidean metric |n − m|, is Zair Ibragimov complete. This is because there are no Cauchy sequences in CSUF Z except for trivial ones like {1, 1, 1,... }, {2, 2, 2,... } and so on. Valuations on Rational So the completion of Z with respect to Euclidean metric is the Numbers same as Z. That is, this completion does not add any new Completions of Rational points to Z. Numbers The p-adic On the other hand, the same integers Z, equipped with the numbers p-adic metric |n − m|p, is NOT complete! For example, the Geometry of p-adic integers following sequence of integers is a nontrivial Cauchy sequence

1, 1 + p, 1 + p + p2, 1 + p + p2 + p3,...

The p-adic integers, denoted by Zp, is the completion of Z with respect to the p-adic ultrametric. The p-adic integers and numbers

Geometry of p-adic The p-adic integers Zp can be identified with the formal series numbers Zair Ibragimov ∞ CSUF X k ak p , where ak ∈ {0, 1, 2,..., p − 1}. Valuations on k=0 Rational Numbers

Completions The p-adic numbers Qp can be identified with the formal series of Rational Numbers ∞ The p-adic X k numbers ak p , where m ∈ Z and ak ∈ {0, 1, 2,..., p − 1}.

Geometry of k=m p-adic integers Problem Show that the sequence of partial sums of each series above is a Cauchy sequence. Use the series representations above to show that Zp = {x ∈ Qp : |x|p ≤ 1}. Decomposition of Z

Geometry of p-adic Our goal is to obtain the p-adic integers as the end space of a numbers metric tree. For each k ∈ {1, 2, 3,..., p} consider the function Zair Ibragimov CSUF fk on Z, defined by fk (z) = pz + (k − 1).

Valuations on Rational Problem Numbers −1 Show that |fk (z) − fk (w)|p = p |z − w|p for all z, w ∈ . Completions Z of Rational Numbers We have the following decompositions of (Z, | − |p): The p-adic numbers = B ∪ B ∪ · · · ∪ B , where B = f ( ). Geometry of Z 1 2 p k k Z p-adic integers Problem

For each k = 1, 2,... p, show that Bk = {pn + (k − 1): n ∈ Z}. Show that Bk is a ball in (Z, | − |p) of radius 1/p. Show that the diameter of Bk is also equal to 1/p. Decomposition of Z (continued)

Geometry of For each fixed n ∈ {1, 2,..., p} we have p-adic numbers p Zair Ibragimov Bn = ∪k=1Bkn, Bkn = fk (Bn) = (fk ◦ fn)(Z). CSUF

Valuations on Problem Rational Numbers −2 −1 Show that diam(Bkn) = p and dist(Bk1n, Bk2n) = p . Show Completions 2 of Rational that each Bkn is a ball in (Z, | − |p) of radius 1/p . Numbers The p-adic For each fixed pair n, m ∈ {1, 2,..., p} we have numbers p Geometry of B = ∪ B , B = f (B ) = (f ◦ f ◦ f )( ). p-adic integers nm k=1 knm knm k nm k n m Z

Problem −3 −2 Show that diam(Bknm) = p and dist(Bk1nm, Bk2nm) = p . 3 Show that each Bknm is a ball in (Z, | − |p) of radius 1/p . The process continues... Collection of balls in (Z, |n − m|p)

Geometry of p-adic numbers Let Zp be the collection of all the balls in (Z, | − |p) described Zair Ibragimov CSUF above. Equip Zp with the distance function hp,

Valuations on Rational diam(A ∪ B) hp(A, B) = 2 log . Numbers pdiam(A) diam(B) Completions of Rational Numbers The p-adic Problem numbers

Geometry of Show that hp is a metric on Zp. p-adic integers Next, we define a metric tree associated with the above decomposition. A X is called a metric tree if each pair of points in it can be joined by a unique arc and this arc is a geodesic segment. Metric tree associated with Zp

Geometry of Consider the set Z as the set of vertices and connect each ball p-adic p numbers B to its children f1(B), f2(B),..., fp(B) by an edge of length Zair Ibragimov CSUF log p. Note that hp(B, fk (B)) = log p.

Valuations on For each pair (B1, B2) of distinct vertices there is a unique Rational Numbers vertex B of smallest diameter containing B1 and B2. Completions of Rational Problem Numbers

The p-adic Let B, B1, B2 be as above. Show that numbers diam(B) = diam(B1 ∪ B2). Geometry of p-adic integers Let γk , k = 1, 2, be the arc connecting B to Bk . Hence length(γk ) = hp(B, Bk ). Then γ = γ1 ∪ γ2 is the unique arc joining B1 and B2. Since

hp(B1, B2) = hp(B, B1) + hp(B, B2) = length(γ), we conclude that γ is a geodesic segment. End space of Tp

Geometry of p-adic The resulting tree, denoted by Tp, is a metric tree. The unit numbers ball Z in Zp is the root of Tp and the pair (Tp, Z) is referred to Zair Ibragimov CSUF as the rooted tree.

Valuations on The end space End(Tp, ) of the pair (Tp, ) is defined to be Rational Z Z Numbers the set of (infinite) geodesic rays emanating from the root Z. Completions of Rational If γ and γ are two such rays, let b(γ , γ ) be their bifurcation Numbers 1 2 1 2

The p-adic point, i.e., the vertex where they split and let l(γ1, γ2) be the numbers length of their common part. Geometry of p-adic integers The distance between γ1 and γ2 is defined to be

−l(γ1,γ2) ρ(γ1, γ2) = e .

We have l(γ1, γ2) = hd (Z, b(γ1, γ2)) = n log p, where n is the degree of separation of b(γ1, γ2) from Z. Zp is the end space of Tp

Geometry of p-adic numbers

Zair Ibragimov We conclude that CSUF

−l(γ1,γ2) −n log p −n Valuations on ρ(γ1, γ2) = e = e = p . Rational Numbers

Completions of Rational Problem Numbers Show that ρ is a metric on End(Fp, Z). The p-adic numbers

Geometry of p-adic integers Theorem (Ibragimov, 2010)  The space End(Fp, Z), ρ is isometric to (Zp, | − |p).