Classification — flashcards

|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.

An absolute value on a field satisfies |x| = 0 exactly when x = 0, |xy| = |x||y|, and |x + y| ≤ |x| + |y|. Ostrowski's theorem is the statement that on ℚ these three conditions have far fewer solutions than one would guess.

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.

The only normed division algebras over ℝ are ℝ, ℂ, ℍ and 𝕆, of dimensions 1, 2, 4 and 8. This is why the ladder stops.

Every elliptic curve over ℚ is modular. Proved for semistable curves by Wiles with Taylor (1995), in general by Breuil, Conrad, Diamond and Taylor (2001). Fermat's Last Theorem is a corollary.

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.

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.

The p-adic L-function and the algebraically defined Iwasawa module generate the same ideal. Proved for ℚ in 1984 — worth contrasting with Birch–Swinnerton-Dyer, its classical analogue, which remains open.

David Hilbert, 1897.

Alexander Ostrowski, 1916.

Adolf Hurwitz, 1898.

Wiles with Taylor; then Breuil, Conrad, Diamond and Taylor, 1995, completed 2001.

Artin and Whaples, 1945.

Kurt Hensel, 1897.

József Kürschák, 1912.

Alexander Ostrowski, 1916.

Barry Mazur and Andrew Wiles, 1984.

Rat.AbsoluteValue.equiv_real_or_padic — Mathlib.NumberTheory.Ostrowski

padicNorm — Mathlib.NumberTheory.Padics.PadicNorm

Climbing the ladder surrenders order, then commutativity, then associativity — and the amounts are not equal. This edge is the same observation as the What breaks edge; it is here because how much is forgotten and what is surrendered are different questions.

Being an integer is what survives every valuation at once — a condition imposed on a larger object, not a starting point that everything else is built up from.

The reals are the completion at the infinite place. Being the ambient number system you happened to grow up with is a fact about pedagogy, not about ℝ.

A geometry is the invariants of a group acting on a space, so choosing Euclidean geometry is choosing one group out of many. Klein's move is this schema applied to geometry wholesale.

The commutative, associative, ordered case is one rung of a short ladder that provably stops at eight dimensions. It is the rung everybody starts on and the least general one.

A geometry ↔ its group of transformations; the theorems of that geometry ↔ the invariants of that group. Euclidean, affine and projective geometry become one family indexed by groups.

Abelian extensions of a number field ↔ quotients of its idele class group. The abelian, fully proved fragment of Langlands.

Quadratic reciprocity is the first visible instance of the pattern class field theory completes for all abelian extensions.

Galois representations ↔ automorphic representations, matched by their L-functions. Proved for elliptic curves over ℚ (modularity, Wiles–Taylor, giving Fermat) and in many other cases; open in general.

Recorded as conjectural: expected, not proved.

Each automorphic form produces an L-function too. Langlands asserts these two supplies of L-functions coincide.

Automorphic forms are best understood as representations of adelic groups — which is what makes functoriality expressible at all.

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.

Ordinary analysis is what you get by taking the one branch we happen to live on. Naming it a branch rather than the default is the whole point of the lens.

Descending ℍ → ℂ means choosing a U(1) inside SU(2) — a whole sphere of equally valid choices. Descending 𝕆 → ℍ offers three. Descending ℂ → ℝ offers one. You noticed the descent is non-uniform before I did.

The first rung of the ladder. Adjoining i buys algebraic closure and costs the ability to compare any two elements — which is why there is no sensible ordering on ℂ.

Doubling again costs ab = ba. The loss is productive rather than merely regrettable: non-commutativity is exactly what lets quaternions encode rotation in three dimensions.

The last rung costs (ab)c = a(bc). Hurwitz says there is nothing further to surrender, which is why the ladder stops at eight dimensions rather than continuing.

REFUTED as proposed. The instinct that a complex-analytic proof of the necklace theorem wouldn't generalize was right; the reason given (a quaternion/division-algebra obstruction) was wrong. The actual obstruction is Adams' Hopf-invariant-one theorem — non-parallelisability of spheres — related territory, wrong mechanism.

Recorded as refuted: this edge was drawn and then withdrawn.

`Mathlib.NumberTheory.Ostrowski` imports `Mathlib.Analysis.AbsoluteValue.Equivalence`, `Mathlib.Analysis.SpecialFunctions.Log.Base`, `Mathlib.Analysis.SpecialFunctions.Pow.Continuity`. Read off Mathlib/NumberTheory/Ostrowski.html. No human asserted this edge.

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

The abelian case of reciprocity, completed in the 1920s–30s: it classifies all abelian extensions of a number field in terms of data inside the field itself.

Extensions of the rationals built from roots of polynomials — where number theory and algebra fuse.

Klein's 1872 reorganisation: a geometry is defined by the group of transformations under which its statements stay true.

Highly symmetric analytic functions on the upper half-plane and its generalisations. The analytic side of Langlands. (Filed under calculus only because an eight-way split leaves no separate slot for analysis or representation theory.)

Every absolute value on ℚ is equivalent to the ordinary one or to a p-adic one. This is the fork itself: it says exactly what the branches are and that there are no others.

ℝ → ℂ → ℍ → 𝕆. Each rung doubles dimension and surrenders a property: order, then commutativity, then associativity. Hurwitz's theorem says it stops at eight.