Resonator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Template:Harvs, is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.

History

Template:Harvs used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Template:Harvtxt. Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form,

$\Delta (q)=q\left(\prod _{n=1}^{\infty }(1-q^{n})\right)^{24}=\sum _{n=1}^{\infty }\tau (n)q^{n},\quad q=e^{2\pi i\tau },$

is a multiplicative function:

$\tau (mn)=\tau (m)\tau (n)\quad {\text{ for }}(m,n)=1.$

The idea goes back to earlier work of Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators.

Mathematical description

Hecke operators can be realised 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

$\sum f(\Lambda ')$

with the sum taken over all the Λ′ that are subgroups of Λ of index n. For example, with n=2 and two dimensions, there are three such Λ′. Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to homotheties, as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.

Another way to express Hecke operators is by means of double cosets in the modular group. In the contemporary adelic approach, this translates to double cosets with respect to some compact subgroups.

Explicit formula

Let M_m 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

$T_{m}f(z)=m^{k-1}\sum _{\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma \backslash M_{m}}(cz+d)^{-k}f\left({\frac {az+b}{cz+d}}\right),$

where z is in the upper half-plane and the normalization constant m^k−1 assures that the image of a form with integer Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form

$T_{m}f(z)=m^{k-1}\sum _{a,d>0,ad=m}{\frac {1}{d^{k}}}\sum _{b{\pmod {d}}}f\left({\frac {az+b}{d}}\right),$

which leads to the formula for the Fourier coefficients of T_mf(z) = ∑ b_nqⁿ in terms of the Fourier coefficients of f(z) = ∑ a_nqⁿ:

$b_{n}=\sum _{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}}.$

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 S_k 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 T_m with eigenvalues λ_m then a_m = λ_ma₁ and a₁ ≠ 0. Hecke eigenforms are normalized so that a₁ = 1, then

$T_{m}f=a_{m}f,\quad a_{m}a_{n}=\sum _{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}},\ m,n\geq 1.$

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 commutative rings. Other, related, mathematical rings are called Hecke algebras, although 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. In the classical elliptic modular form theory, the Hecke operators T_n 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).

See also

• Eichler–Shimura congruence relation

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• Jean-Pierre Serre, A course in arithmetic.

• Don Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms, Universitext, Springer, 2008 ISBN 978-3-540-74117-6