Lifting

A theorem recovers fuller information from residual information under extra conditions.

Start with a concrete example

The remainder is 2 after division by 3. Was the original number 2, 5, 8 or 11? The remainder alone cannot tell us. Add the condition “between 6 and 10”.

Move the number or condition

The remainder alone cannot choose. There are more candidates beyond this list too.

How this example differs from the definition

The interval is a toy condition, not the hypotheses of a modularity-lifting theorem. The real theorem concerns representations and requires its own full hypotheses.

A small check: try the idea

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

If the remainder after division by 3 is 2, must the original number be 8?

Read the answer and reasoning

No; 2, 5, 8 and 11 are all possible. Turn the extra condition off: all four displayed numbers leave 2. The toy condition 6 < n < 10 singles out 8; a real lifting theorem needs its own hypotheses.

Build the idea in three steps

Reducing information can make a problem simpler, but it also forgets information.
A lifting theorem restores a richer object under carefully stated conditions. There is no automatic inverse to taking remainders.
In these chapters residual modularity, irreducibility, semistability and level restrictions prepare inputs to a modularity-lifting theorem.

A small example

Knowing an integer is 2 modulo 5 does not tell you whether it was 2, 7 or 12. Extra structure is essential for lifting.

Use in Lean · original excerpt

FLT · aa2d8b3 · S_WeierstrassCurve_modularity_of_semistableModel.lean

174    exact isModular_map_of_mlc1175      (liftingAtConductor_three_of_level_hns 3 rfl W hΔ hW h3 M₃ hres₃ hns₃ hcube₃)

How to read this Lean

This is a theorem application, not a definition of a generic inverse operation. The arguments pass the collected evidence to liftingAtConductor_three_of_level_hns.

Original source ↗ · L174–175 Open the complete saved source

Where this idea is used

9. Two doors to the concert 10. A three-note audition 11. Borrow a better passport 12. The workshops agree 13. Pip’s impossible quilt

Lifting

Start with a concrete example

A small check: try the idea

Build the idea in three steps

A small example

Use in Lean · original excerpt

How to read this Lean

Where this idea is used

Mika’s help desk

One thing at a time

In simpler words

A small clue