Local–global principle — flashcards

Every card the map can derive about Local–global principle — definitions, statements, attribution, and the edges it sits on.

9 cards. Click a question to reveal the answer — this uses no scripting, so the deck works with JavaScript disabled and prints as a plain question-and-answer list.

Theorems 2
Open and counterexample 1
Names and dates 3
Connections 2
Concepts 1

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Theorems 2

State: Hasse–Minkowski theorem.

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.

derived from st/hasse-minkowski by the statement rule

State: The Selmer group is computable.

In 0 → E(ℚ)/2E(ℚ) → Sel₂(E) → Ш(E)[2] → 0 the middle term is assembled from local conditions at each place and so is mechanically computable; Ш is precisely the local–global failure that remains.

derived from st/selmer-descent by the statement rule

Open and counterexample 1

What does Selmer's cubic show, and why does it matter?

3x³ + 4y³ + 5z³ = 0 is solvable over ℝ and over every ℚₚ, and has no nontrivial rational solution. Found in 1951; it is why the local–global principle is a phenomenon rather than a theorem.

derived from st/selmer-cubic by the statement rule

Names and dates 3

Hasse–Minkowski theorem — who, and when?

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

derived from st/hasse-minkowski by the attribution rule

Selmer's cubic — who, and when?

Ernst Selmer, 1951.

derived from st/selmer-cubic by the attribution rule

The Selmer group is computable — who, and when?

Ernst Selmer, 1951.

derived from st/selmer-descent by the attribution rule

Connections 2

Adeles & ideles and Local–global principle look like one thing. What separates them? [Forks]

You may study a problem at a single completion, or assemble every place at once into the adeles. Neither is the default; the product formula says no place is privileged, and choosing one is a choice.

derived from forks:adeles→local-global by the connection rule

What breaks on the way from Local–global principle to Local data determining global? [What breaks]

Hasse–Minkowski gives local-implies-global for quadratic forms. Raise the degree by one and Selmer's cubic destroys it. The cost of a single extra degree is the entire principle — and the Tate–Shafarevich group exists to measure exactly how much was lost.

derived from what-breaks:local-global→prop/local-global by the connection rule

Concepts 1

What is Local–global principle, in one sentence?

Solve a problem over the reals and over every ℚₚ, and ask whether that forces a solution over ℚ. It holds for quadratic forms and fails for cubics — the failure is where the interesting objects live.

derived from local-global by the concept rule