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Semistability

A condition controlling how a curve behaves when reduced at each prime.

Start with a concrete example

From a curve we calculate two numbers, called Δ and c₄. To practice the check, pretend they are 30 and 7. The primes dividing 30 are 2, 3 and 5. None divides 7. Now change the second number.

Δ = 30, c₄ = 72does not divide c₄3does not divide c₄5does not divide c₄

This pair passes the illustrated divisibility condition.

How this example differs from the definition

The two numbers here are chosen for a divisibility exercise, not asserted to come from an actual curve. The real curve supplies both numbers and all relevant primes must be checked.

A small check: try the idea

Compare the numbers in the question with the diagram you are moving. You can try again.

In the toy test Δ=30, c₄=6, do the required divisibility conditions pass?

Read the answer and reasoning

No; 2 and 3 divide both. Choose c₄=6. We test primes dividing 30: 2,3,5. Both 2 and 3 also divide 6, so the condition fails for this toy pair.

Build the idea in three steps

  1. Reducing coefficients modulo a prime can change a curve’s behavior. Some primes give bad reduction.
  2. The discriminant Δ and invariant c₄ are quantities computed from the curve’s coefficients. This repository tests which primes divide them.
  3. For every prime dividing Δ, semistability requires it not to divide c₄. This exact condition supports the later theorems.

A small example

The passport is not a visual smoothness test. It is evidence for the prime-by-prime divisibility statement shown in Lean.

Lean definition · original excerpt

FLT · aa2d8b3 · Def_FLTPrelim_Modularity.lean

def IsSemistableModel (W : WeierstrassCurve ℤ) : Prop :=  ∀ p : ℕ, p.Prime → (p : ℤ) ∣ W.Δ → ¬ (p : ℤ) ∣ W.c₄

How to read this Lean

p.Prime requires a prime; ∣ means divides; ¬ negates a statement. Read the arrows as “if p is prime and divides Δ, then it does not divide c₄”.

Original source ↗ · L85–86Open the complete saved source

Where this idea is used

9. Two doors to the concert10. A three-note audition11. Borrow a better passport12. The workshops agree13. Pip’s impossible quilt14. The shrinking reservation16. The room with no room
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Semistability

One thing at a time

Read the three explanation steps first. The Lean source is there for a closer look afterwards; you do not have to read it first.

In simpler words

A condition controlling how a curve behaves when reduced at each prime.

Earlier ideas are listed under “Start with these ideas” on this page.

A small clue

The passport is not a visual smoothness test. It is evidence for the prime-by-prime divisibility statement shown in Lean.

Close this panel to return to the exact place you were reading. These are prepared hints for this page.

Start with the first small story →