Hecke algebra

In mathematics, the Hecke algebra is the algebra generated by Hecke operators.

For other mathematical rings called Hecke algebras, see Hecke algebra (disambiguation).

Properties

The algebra is a commutative ring.[1]

In the classical elliptic modular form theory, the Hecke operators Tn with n coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the inverse of the Hecke polynomial, a quadratic polynomial in p−s. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of τ(n).

gollark: Duckduckgoing it just turns up a lot of information on compiling *everything* like that, which is slow.
gollark: I thought there was a way to do this but I forgot it; can you compile a single dependency with a higher optimization level?
gollark: Oh, never mind, found it.
gollark: Thanks. Apparently that works. Is there a way to *cancel* that task from the function which spawns it?
gollark: I think I'm missing something then. It says```rusterror[E0373]: async block may outlive the current function, but it borrows `ws`, which is owned by the current function --> src/connection.rs:40:23 |40 | task::spawn(async { | _______________________^41 | | let mut interval = stream::interval(Duration::from_secs(10));42 | | while let Some(_) = interval.next().await {43 | | ws.send_string("Hi".to_string()); | | -- `ws` is borrowed here44 | | }45 | | }); | |_____^ may outlive borrowed value `ws````

References
  1. Serre 1973, Ch. VII, § 5. Corollary 2.


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.