Hecke operator
In
History
is a multiplicative function:
The idea goes back to earlier work of
Mathematical description
Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer n some function f(Λ) defined on the lattices of fixed rank to
with the sum taken over all the Λ′ that are
Another way to express Hecke operators is by means of
Explicit formula
Let Mm be the set of 2×2 integral matrices with determinant m and Γ = M1 be the full modular group SL(2, Z). Given a modular form f(z) of weight k, the mth Hecke operator acts by the formula
where z is in the upper half-plane and the normalization constant mk−1 assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form
which leads to the formula for the Fourier coefficients of Tm(f(z)) = Σ bnqn in terms of the Fourier coefficients of f(z) = Σ anqn:
One can see from this explicit formula that Hecke operators with different indices commute and that if a0 = 0 then b0 = 0, so the subspace Sk of cusp forms of weight k is preserved by the Hecke operators. If a (non-zero) cusp form f is a simultaneous eigenform of all Hecke operators Tm with eigenvalues λm then am = λma1 and a1 ≠ 0. Hecke eigenforms are normalized so that a1 = 1, then
Thus for normalized cuspidal Hecke eigenforms of integer weight, their Fourier coefficients coincide with their Hecke eigenvalues.
Hecke algebras
Algebras of Hecke operators are called "Hecke algebras", and are
Other related mathematical rings are also called "Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the group algebras of braid groups. The presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.
See also
- Eichler–Shimura congruence relation
- Hecke algebra
- Abstract algebra
- Wiles's proof of Fermat's Last Theorem
References
- ISBN 978-0-387-97127-8(See chapter 8.)
- "Hecke operator", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Hecke, E. (1937a), "Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I.", Zbl 0015.40202
- Hecke, E. (1937b), "Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II.", Zbl 0016.35503
- JFM 46.0605.01
- Jean-Pierre Serre, A course in arithmetic.
- ISBN 978-3-540-74117-6