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Hecke algebras

Collect certain operators on cusp forms and the arithmetic they encode.

Start with a concrete example

Starting with “double a number” and “triple a number”, we can combine operations by addition and composition. A Hecke algebra likewise starts from specified mathematical operators; it is not the collection of every possible action.

A small check: try the idea

Think with the concrete example above. You can try again.

Is every possible action on a function automatically a Hecke operator?

Read the answer and reasoning

No; the mathematical operators are specified. The example begins with chosen operations such as doubling and tripling. Likewise, the actual Hecke algebra uses its defined operators, not arbitrary actions.

Build the idea in three steps

  1. An operator takes a form to another form; linear operators preserve addition and scalar multiplication.
  2. Hecke operators are particular arithmetic operators on spaces of forms. Their eigenvalues connect to coefficient data.
  3. The Hecke algebra collects specified operators and the combinations generated by addition and multiplication. It is the second workshop in the story.

A small example

Do not identify “all operators” with “the Hecke algebra”. The definition selects particular generators first.

Lean definition · original excerpt

FLT · aa2d8b3 · Def_CuspForm_HeckeAlgebra.lean

def heckeGenerators : Set (Module.End ℂ (CuspForm (CongruenceSubgroup.Gamma0 N) k)) :=  {T | ∃ (ℓ : ℕ) (hℓ : ℓ.Prime) (hℓN : ¬ ℓ ∣ N), ℓ ∉ S ∧ T = heckeTLin k hℓ hℓN} ∪    {U | ∃ (q : ℕ) (hqN : q ∣ N), q.Prime ∧ q ∉ S ∧ U = heckeULin k hqN} def heckeAlgebra : Subalgebra ℤ (Module.End ℂ (CuspForm (CongruenceSubgroup.Gamma0 N) k)) :=  Algebra.adjoin ℤ (heckeGenerators N k S)

How to read this Lean

heckeGenerators selects the T and U operators with prime and level conditions. Algebra.adjoin builds the subalgebra generated by that set.

Original source ↗ · L14–19Open the complete saved source

Where this idea is used

12. The workshops agree13. Pip’s impossible quilt
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Hecke algebras

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

Collect certain operators on cusp forms and the arithmetic they encode.

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

A small clue

Do not identify “all operators” with “the Hecke algebra”. The definition selects particular generators first.

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 →