Elliptic curves — flashcards

Every card the map can derive about Elliptic curves — 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 3
Open and counterexample 1
Names and dates 4
Concepts 1

This deck is generated. No card was written for the deck: each one is derived from a node in the registry by a stated rule, and names its source underneath the answer. Correct the statement and the card follows.

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Theorems 3

State: Mordell–Weil theorem.

The rational points of an elliptic curve over a number field form a finitely generated abelian group — so infinitely many points are always generated by finitely many.

derived from st/mordell-weil 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

State: Chabauty's method.

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.

derived from st/chabauty 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 4

Selmer's cubic — who, and when?

Ernst Selmer, 1951.

derived from st/selmer-cubic by the attribution rule

Mordell–Weil theorem — who, and when?

Louis Mordell, then André Weil, 1922 (Mordell), 1929 (Weil).

derived from st/mordell-weil by the attribution rule

The Selmer group is computable — who, and when?

Ernst Selmer, 1951.

derived from st/selmer-descent by the attribution rule

Chabauty's method — who, and when?

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

derived from st/chabauty by the attribution rule

Concepts 1

What is Elliptic curves, in one sentence?

Smooth genus-one curves with a marked point, whose points form a group. Filed under geometry because it is a curve, studied almost entirely for its arithmetic — the clearest single case of this map's eight-way split failing to hold an object.

derived from elliptic-curves by the concept rule