p-adic numbers — flashcards

Write a nonzero rational x as p^n · a/b with p dividing neither a nor b; then v_p(x) = n, and v_p(0) = +∞ by convention.

|x|_p = p^−v_p(x) for x ≠ 0, and |0|_p = 0. Divisibility by a high power of p is what it means to be small.

|x + y|_p ≤ max(|x|_p, |y|_p), strictly stronger than the ordinary triangle inequality, with equality whenever |x|_p ≠ |y|_p.

Every element of ℚ_p is a series in powers of p extending infinitely toward *higher* powers, with only finitely many negative ones — the mirror image of a decimal. In ℚ₅, −1 = 4 + 4·5 + 4·5² + ⋯, which is the honest version of the schoolyard “…999 = −1”.

ℤ_p = { x ∈ ℚ_p : |x|_p ≤ 1 }, equivalently the inverse limit of ℤ/pⁿℤ. A local ring with unique maximal ideal pℤ_p and residue field 𝔽_p.

For a, b in ℚ*, the product of the local Hilbert symbols over all places equals 1. Reciprocity is this statement, and it is why the completions are not independent.

Every nontrivial absolute value on ℚ is equivalent either to the ordinary absolute value or to a p-adic one for exactly one prime p.

If f(a) ≡ 0 mod p and f'(a) ≢ 0 mod p, then f has a unique p-adic root congruent to a. It is Newton's method with the ultrametric inequality doing the work that real-analytic estimates do over ℝ.

A quadratic form over ℚ has a nontrivial rational zero if and only if it has one over ℝ and over every ℚₚ. The first major local–global principle.

For nonzero rational x, the ordinary absolute value times the product of all p-adic ones equals 1. Every place is accounted for, and no single one is privileged.

Each nonzero residue mod p lifts uniquely to a (p−1)st root of unity in ℤ_p, giving a canonical multiplicative section of ℤ_p× → 𝔽_p×. The residue field sits inside the ring, multiplicatively but not additively.

Nearness in |·|_p and nearness in |·|_∞ are unrelated: by Ostrowski the absolute values are inequivalent, so neither induces any information about the other. Asking whether a p-adic number is larger than a real one has no content.

Because |x + y|_p = max(|x|_p, |y|_p) exactly when the two differ, among any three ultrametric distances the two largest are equal. The metric intuition imported from ℝ fails at the first triangle.

Balls in ℚ_p are simultaneously open and closed, so ℚ_p is totally disconnected — and yet not discrete, since ℤ is dense in ℤ_p.

If c ∈ B(a, r) then B(c, r) = B(a, r). Two balls are therefore either disjoint or nested; partial overlap is impossible.

ℤ_p is a closed subset of the product of the finite discrete spaces ℤ/pⁿℤ, hence compact by Tychonoff — or directly, by pigeonholing digits. ℚ_p is only locally compact.

A series converges in ℚ_p if and only if its terms tend to zero. The condition that is merely necessary over ℝ is sufficient here, and the harmonic series has no p-adic analogue of its failure.

exp_p converges for v_p(x) > 1/(p−1) and log_p(1+x) for v_p(x) ≥ 1; on their common domain they invert one another and give (1 + pℤ_p, ×) ≅ (pℤ_p, +) as topological groups. p = 2 needs the smaller disc.

For odd p, ℤ_p× ≅ μ_{p−1} × (1 + pℤ_p) ≅ ℤ/(p−1)ℤ × ℤ_p — Teichmüller roots of unity times the one-units. For p = 2 it is ℤ/2ℤ × ℤ_2, generated by −1 and 1 + 4ℤ_2.

v_p(n!) = Σ_{i≥1} ⌊n/pⁱ⌋, which grows like n/(p−1). So Σ n! converges p-adically for every prime while diverging in every real sense.

Any field carrying an absolute value has a completion, constructed exactly as ℝ is built from ℚ. This is what puts ℚ_p on the same footing as ℝ rather than treating it as an exotic construction.

If |f(a)|_p < |f′(a)|_p² then f has a unique root α with |α − a|_p < |f′(a)|_p. Dropping the requirement that a be a simple root mod p is what makes the lemma usable in practice.

The zeta function of a variety over a finite field is a rational function — the first of the Weil conjectures, and proved by p-adic analysis rather than by the geometry the conjectures were about.

Kummer's congruences between Bernoulli numbers are the shadow of a single p-adic analytic function interpolating ζ(1−n). A 19th-century family of coincidences turns out to be one object.

st/tate-thesis reproves Hecke L-function analytic continuation via Fourier analysis on the adeles. At a single p-adic place, the local piece of that machinery — Tate's local zeta integral Z(f,s) = ∫_{Q_p*} f(x)|x|_p^s d*x — is exactly the p-adic analogue of the (archimedean) Mellin transform, using the multiplicative Haar measure d*x in place of dx/x. For f the indicator of Z_p, it sums to (1−p⁻¹)/(1−p^{-(s+1)}), the local Euler factor itself — so the Euler factor at p is literally a Mellin transform value, not just an analogue of one by loose resemblance.

For a polynomial map over Q_p, whether iteration is chaotic is a question about reduction mod p, not about the field being non-archimedean per se. Under good reduction (e.g. f(x)=4x(1-x) at odd p), the multiplier stays a p-adic unit away from the critical point, so |f(x)-f(x')|_p = |x-x'|_p exactly at every step — valuations are preserved, not amplified, and the Julia set is empty (Rivera-Letelier; Benedetto). Under bad reduction (e.g. f(z)=z²+c with |c|_p>1), the identity f(z)-f(w)=(z-w)(z+w) instead gives genuine expansion by |c|_p^{1/2} per step on the Julia set: a repelling fixed point exists there, gaps between nearby points grow by an exact integer factor each iteration, giving a positive Lyapunov exponent and dynamics conjugate to the full 2-shift. So 'chaos is archimedean' is false as a general claim — the real dividing line is good vs. bad reduction, a condition on the map's coefficients relative to p, independent of which absolute value is in play.

Two constructions can give Cantor sets with the same Hausdorff dimension over R while behaving completely differently once the absolute value changes. A digit-restriction set (base-p digits confined to a subset of size d) has dimension log(d)/log(p) at every place where the construction makes sense — purely geometric, place-independent, verified matching a middle-thirds-style real Cantor set exactly. A Julia set's dimension is not: under good reduction (st/padic-reduction-and-chaos) it collapses to 0 in Q_p even where the real Julia set has dimension strictly between 0 and 1, and under bad reduction it takes its own value, log(2)/log(r) for expansion factor r=|c|_p^{1/2}, moving in the opposite direction from the real dimension as a parameter varies — with neither the sum nor the product of the two staying constant, so there is no product-formula analogue for dimension the way there is for absolute values themselves.

When the Mordell–Weil rank is less than the genus, p-adic integration bounds the number of rational points on a curve. Analysis over ℚ_p answering a question ℝ cannot touch.

For each degree d, Artin's conjecture on forms over ℚ_p holds for all but finitely many p — proved by model theory, by showing ℚ_p and 𝔽_p((t)) become elementarily equivalent as p grows.

A tilting equivalence relating characteristic 0 and characteristic p geometry, which turned a long list of separate p-adic comparison results into instances of one construction.

For a = 2^m·u, b = 2^n·v with u, v odd 2-adic units, (a,b)₂ = (−1)^(ε(u)ε(v) + mω(v) + nω(u)), where ε(u) = (u−1)/2 mod 2 and ω(u) = (u²−1)/8 mod 2 — both fixed-depth congruences (u mod 4, u mod 8) independent of the size of u.

By Hensel's lemma, whether an odd 2-adic unit u is a square depends only on u mod 8 — a fixed-width digit-read, however large u is. No analogous fixed k exists at an odd prime q: whether u is a quadratic residue mod q is not determined by any bounded number of low bits of u. Checked directly: u=3 vs u=11 (agreeing mod 8) disagree on 339/666 tested primes q; u=7 vs u=71 (agreeing mod 64) disagree on 500/1004 — matching more shared bits does not shrink the disagreement rate toward 0. This is the precise sense in which quadratic reciprocity's supplementary laws are free lookups while the reciprocity step itself is not, and it is what st/hilbert-product's 2-adic term is actually doing.

David Hilbert, 1897.

Alexander Ostrowski, 1916.

Kurt Hensel, 1897.

Hermann Minkowski, then Helmut Hasse, 1890 (Minkowski), 1923 (Hasse).

Artin and Whaples, 1945.

Kurt Hensel, 1897.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Alexander Ostrowski, 1916.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

József Kürschák, 1912.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Bernard Dwork, 1960.

Kubota and Leopoldt, 1964.

Claude Chabauty, then Robert Coleman, 1941 (Chabauty), 1985 (Coleman).

James Ax and Simon Kochen, 1965.

Peter Scholze, 2012.

Rat.AbsoluteValue.equiv_real_or_padic — Mathlib.NumberTheory.Ostrowski

padicValRat — Mathlib.NumberTheory.Padics.PadicVal.Basic

padicNorm — Mathlib.NumberTheory.Padics.PadicNorm

padicNorm.nonarchimedean — Mathlib.NumberTheory.Padics.PadicNorm

PadicInt — Mathlib.NumberTheory.Padics.PadicIntegers

IsUltrametricDist.dist_eq_max_of_dist_ne_dist — Mathlib.Topology.MetricSpace.Ultra.Basic

IsUltrametricDist.isClopen_ball — Mathlib.Topology.MetricSpace.Ultra.Basic

IsUltrametricDist.ball_eq_of_mem — Mathlib.Topology.MetricSpace.Ultra.Basic

PadicInt.compactSpace — Mathlib.NumberTheory.Padics.ProperSpace

sub_one_mul_padicValNat_factorial — Mathlib.NumberTheory.Padics.PadicVal.Basic

hensels_lemma — Mathlib.NumberTheory.Padics.Hensel

Two representations that each throw away the same thing. Neither lets you read off how big a number is from a prefix of its expansion — the ordering information is simply not in the encoding.

Reciprocity is the statement that the local Hilbert symbols over every completion multiply to 1. So the p-adic worlds are not independent — fixing all but one determines the last. This was your own question: whether quadratic reciprocity means the completions are coupled. It does.

Ostrowski's theorem says the branches are exactly ℝ and one ℚₚ per prime. This is the cleanest fork in mathematics: infinitely many consistent completions of the same starting object, and no principled reason to privilege the real one.

The alphabet size is the number of inverse branches of the map. Choosing base p and choosing to work p-adically are versions of the same choice, differing in how the refinement is oriented — disjoint balls rather than intervals sharing boundaries.

`Mathlib.NumberTheory.Ostrowski` imports `Mathlib.NumberTheory.Padics.PadicNorm`. Read off Mathlib/NumberTheory/Ostrowski.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.PadicVal.Basic` imports `Mathlib.NumberTheory.Divisors`. Read off Mathlib/NumberTheory/Padics/PadicVal/Basic.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.PadicVal.Basic` imports `Mathlib.Data.Nat.MaxPowDiv`, `Mathlib.Data.Nat.Multiplicity`, `Mathlib.Data.Nat.Prime.Int`. Read off Mathlib/NumberTheory/Padics/PadicVal/Basic.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Algebra.Polynomial.Identities`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Analysis.SpecificLimits.Basic`, `Mathlib.Analysis.Normed.Field.ProperSpace`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Topology.Algebra.Polynomial`, `Mathlib.Topology.MetricSpace.CauSeqFilter`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

For each prime p, a completion of the rationals in which closeness means divisibility by high powers of p. Non-Archimedean: every triangle is isosceles, and every point of a disc is its centre.