Hedge operator

In mathematics , Hecke operators mean certain linear operators on the vector space of the entire module forms . These operators were introduced by Erich Hecke in 1937. They get their meaning from the fact that certain module forms are simultaneous eigenfunctions of all Hecke operators and thus conclusions can be drawn about the properties of the Fourier coefficients of these functions. These module forms are also called eigen modes .

The Hecke operators form an algebra called Hecke algebra (the name is, however, also used for other algebras in different areas of mathematics, some of which are only distant and not immediately apparent from the definition) and are a commutative ring.

definition

Let it be the vector space of all modular forms with weight k that transform under the module group . ${\displaystyle M_{k}}$ ${\displaystyle \Gamma =SL_{2}(\mathbb {Z} )}$

A hedge operator is a linear mapping ${\displaystyle T_{n}:M_{k}\rightarrow M_{k},n\in \mathbb {N} ,}$

${\displaystyle (T_{n}f)(\tau )=n^{k-1}\sum _{d|n}d^{-k}\sum _{b=0}^{d-1}f\left({\frac {n\tau +bd}{d^{2}}}\right).}$

It is from the upper half-plane ( ). ${\displaystyle \tau }$ ${\displaystyle \operatorname {Im} \tau >0}$

For prime numbers p this is reduced to

${\displaystyle (T_{p}f)(\tau )=p^{k-1}f(p\tau )+{\frac {1}{p}}\sum _{b=0}^{p-1}f\left({\frac {\tau +b}{p}}\right).}$

An equivalent definition describes the effect of Hecke operators as a kind of averaging over elements of the general linear group of integer 2 × 2 matrices (determinant m) modulo the module group (equal , determinant 1): ${\displaystyle \left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)}$ ${\displaystyle M_{m}}$ ${\displaystyle \Gamma }$ ${\displaystyle M_{1}}$

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

with a modular form of degree k . The previous definition seen from this apparent when one considers that the sum of a legal representative system is executed, and this is given by the integer 2 × 2 matrices having determinant , , and , the modulo defined. The number of elements in the legal representative system is equal to the sum of the divisors of . Legal representative system refers to the fact that one considered (with ) the legal multiplication of the effect of in . ${\displaystyle f(z)}$ ${\displaystyle \Gamma \backslash M_{m}}$ ${\displaystyle \left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)}$ ${\displaystyle ad=m}$ ${\displaystyle d>0}$ ${\displaystyle c=0}$ ${\displaystyle b}$ ${\displaystyle d}$ ${\displaystyle d}$ ${\displaystyle m}$ ${\displaystyle \Gamma }$ ${\displaystyle M_{m}}$ ${\displaystyle \Gamma M_{m}=M_{m}}$

An even more general definition of the Hecke operator is given, for example, in Serre, A course in arithmetic , and uses the connection of module functions with grids in the complex plane and is based on the above definition of averaging as the sum of subgrids of a grid of rank n. Hecke operators are mappings in the space of the modular forms (which are assigned to certain grids) when a transition is made from the grid to a sub-grid. ${\displaystyle T_{n}}$

Properties and uses

Hecke operators commute with one another and it applies in the case that the greatest common divisor of m and n is 1 ( ). In this case the Hecke operator is a multiplicative function in number theory. ${\displaystyle T_{m}\,T_{n}=T_{mn}}$ ${\displaystyle ggT(m,n)=1}$

The Hecke operators map themselves, i.e. H. is again an integer k module form to the weight, in particular they form tip shapes , d. H. Module shapes with a zero at , again on peak shapes (for them applies to the zeroth Fourier coefficient ). ${\displaystyle M_{k}}$ ${\displaystyle T_{n}f}$ ${\displaystyle \tau =\infty }$ ${\displaystyle \alpha _{f}(0)=0}$

${\displaystyle f\in M_{k}}$has the Fourier expansion . Then has a Fourier expansion ${\displaystyle f(\tau )=\sum \limits _{m=0}^{\infty }\alpha _{f}(m)e^{2\pi im\tau }}$ ${\displaystyle T_{n}f}$

${\displaystyle (T_{n}f)(\tau )=\sum \limits _{m=0}^{\infty }\gamma _{n}(m)e^{2\pi im\tau }}$ With

${\displaystyle \gamma _{n}(m)=\sum \limits _{d>0,d|(n,m)}d^{k-1}\alpha _{f}\left({\frac {mn}{d^{2}}}\right).}$

The function f is called a simultaneous eigenmode (Hecke eigenmode) if f is an eigenmode for all Hecke operators, in this case the eigenvalues ​​can be normalized so that the first eigenvalue is equal to 1:

${\displaystyle \lambda _{f}(n)=\left.{\frac {\alpha _{f}(n)}{\alpha _{f}(1)}}\right.}$.

and it applies:

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

with . This means that in the case of a Hecke eigenmode, the Fourier coefficients are given as eigenvalues ​​of the Hecke operators and they are thus clearly defined by the Hecke operators. Such a Hecke eigenform exists because the Hecke operators commute with one another. ${\displaystyle \lambda _{1}=1}$

The vector space of the tip shapes (which can be made into a Hilbert space via the Petersson scalar product ) even has a basis of simultaneous eigenfunctions to the operator ${\displaystyle T_{n}}$

If, for example, one chooses the discriminant , which, apart from a constant factor, is the only tip shape of weight 12: ${\displaystyle \Delta }$

${\displaystyle T_{n}\Delta =\tau (n)\cdot \Delta }$ for all ${\displaystyle n\in \mathbb {N} }$

and for its Fourier coefficients , the Ramanujan tau function : ${\displaystyle \tau (n)}$

${\displaystyle \tau (m)\tau (n)=\sum \limits _{d|(m,n)}d^{11}\tau \left({\frac {mn}{d^{2}}}\right).}$

Especially for relatively prime m, n is therefore , d. H. the number theoretic function is multiplicative. ${\displaystyle \tau (m)\tau (n)=\tau (mn)}$ ${\displaystyle \tau (n)}$

The only non-peak forms that are simultaneous eigenmodes for all Hecke operators are the normalized Eisenstein series

${\displaystyle f(\tau )=\left.{\frac {(2k-1)!}{2(2\pi i)^{2k}}}G_{2k}(\tau )\right..}$

For the Fourier coefficients of the Eisenstein series we get:

${\displaystyle \sigma _{2k-1}(n)\sigma _{2k-1}(m)=\sum \limits _{d|(m,n)}d^{2k-1}\sigma _{2k-1}\left({\frac {mn}{d^{2}}}\right)}$

and for coprime m, n this is reduced again to , d. H. the number theoretic function is also multiplicative. ${\displaystyle \sigma _{2k-1}(n)\sigma _{2k-1}(m)=\sigma _{2k-1}(mn)}$ ${\displaystyle \sigma _{2k-1}}$

Hecke operators have many other uses in number theory. The Eichler-Shimura trace formula (according to Martin Eichler , Atle Selberg ), whereby trace means the sum of the eigenvalues ​​of the action of Hecke operators in the space of the tip shapes, was used by Eichler and Selberg for relationships between the Hurwitz class numbers of binary quadratic Derive forms of negative discriminants. Adolf Hurwitz first demonstrated such class number relationships in 1885, which is why they are named after him. Hedge eigenmodes also play a central role in the Serre conjecture .

Atkin-Lehner theory

The space of the tip forms is a Hilbert space with respect to the Petersson dot product (it has an orthonormal system as a basis) and it is often useful to find a basis of simultaneous eigenmodes of the Hecke operators, which is not possible for all spaces of modular forms (here module forms for congruence subgroups are also considered, in the introduction the full module group was considered). In 1970, however, AOL Atkin and Joseph Lehner developed for modular forms of (later also expanded to other transformation groups), by simultaneously considering the spaces for different levels N, a possibility to achieve that for the subspace of the so-called new forms (primitive forms) ( Atkin-Lehner theory). Forms orthogonal to these new forms are called old forms. ${\displaystyle \Gamma _{0}(N)}$

For example, the congruence subgroup of the module group is considered:

${\displaystyle \Gamma _{0}(N)=\left\{{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} ):c\equiv 0{\pmod {N}}\right\}}$

and in addition the nested subgroups that result when considered where is a divisor of . The space of the top shapes to is then a subspace of the space of the top shapes of with the inclusion picture: ${\displaystyle \Gamma _{0}(M)}$ ${\displaystyle M}$ ${\displaystyle N=M\cdot h}$ ${\displaystyle \Gamma _{0}(M)}$ ${\displaystyle \Gamma _{0}(N)}$

${\displaystyle S_{k}(\Gamma _{0}(M))\times S_{k}(\Gamma _{0}(M))\to S_{k}(\Gamma _{0}(N)\,\,\colon \,\,(f(z),g(z))\to f(z)+g(z^{h})}$

All module forms of level N, which result from this inclusion mapping from module forms of levels , where p runs through all prime numbers that divide N, are called the old form. Reforms are the orthogonal complement to this with respect to the Petersson scalar product. These are sometimes also called primitive modular forms. ${\displaystyle {\frac {N}{p}}}$

In the Hilbert space (with Petersson scalar product as inner product) of the associated tip shapes (for a certain secondary type and weight) the Hecke operators for p, which do not share the level N, are self adjoint . This is especially true for the new forms, which are also mapped to themselves under the operation of these Hecke operators (just like the old forms). So one can build an orthogonal basis with respect to the Petersson scalar product of simultaneous eigenforms for all Hecke operators in the space of the newforms. However, if there are too many old forms, this cannot be extended to the entire space of the modular forms. In the case of N = 1, there are no prime divisors and there are no old forms and thus a basis of simultaneous eigenmodes for the whole space. In the case of k = 2 there is no non-vanishing tip shape for the full module group (N = 1) and thus only new shapes for N = p, so here the existence of a basis of eigenforms for the whole space is ensured. ${\displaystyle T_{p}}$

Relationship between module shapes and Dirichlet rows (Hecke L rows)

Let be a modular form of weight 2 k (with ) and Fourier coefficients : ${\displaystyle f(z)}$ ${\displaystyle k>0}$ ${\displaystyle a(n)}$

${\displaystyle f(z)=\sum _{n=1}^{\infty }a(n)q^{n}}$

the eigenfunction of all Hecke operators is ( ): ${\displaystyle T_{n}}$ ${\displaystyle n\geq 1}$

${\displaystyle T_{n}f(z)=\lambda (n)T(n)}$

you can normalize them on and show that for . ${\displaystyle a(1)=1}$ ${\displaystyle a(n)=\lambda (n)}$ ${\displaystyle n>1}$

Two normalized module functions with the same eigenvalues ​​of the Hecke operators are identical. The following applies to the Fourier coefficients : ${\displaystyle a(n)}$

${\displaystyle a(n)\,a(m)=a(n\,m)}$ if gcd (n, m) = 1.

${\displaystyle a(p)\,a(p^{m})=a(p^{m+1})+p^{2k-1}\,a(p^{m-1})}$ (for prime p)

since the Fourier coefficients satisfy the same identities as the Hecke operators.

Hecke recognized that an L-series ( Dirichlet series ) can be formed with the Fourier coefficients of a modular form:

${\displaystyle L_{f}(s)=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}}$

with complex numbers , it converges absolutely for and, according to Hecke, can be analytically continued to a meromorphic function in the entire complex plane (if the modular shape is a peak shape, the continuation is even holomorphic). ${\displaystyle s}$ ${\displaystyle Re(s)>2k}$

It fulfills an Euler product formula:

${\displaystyle L_{f}(s)=\prod _{p\ {\rm {prim}}}{\frac {1}{1-a(p)\,p^{-s}+p^{2k-1-2s}}}}$

This follows from the product formulas given above for the Fourier coefficients (and vice versa from the Euler product formula the coefficient product formula).

The function formed with the gamma function

${\displaystyle X_{f}(s)={\frac {1}{{(2\pi )}^{s}}}\Gamma (s)L_{f}(s)}$

satisfies a functional equation:

${\displaystyle X_{f}(s)={(-1)}^{k}\,X_{f}(2k-s)}$

The behavior of the module shape on inversion is used to prove this

${\displaystyle f(-{\frac {1}{z}})=z^{2k}f(z)}$

and the Mellin transform of the modular form.

Hecke proved that every Dirichlet series that has a functional equation and Euler product development of the above form and fulfills some regularity and growth conditions can be derived from a modular form with a weight of 2k. In addition, this modular form is a simultaneous eigenfunction of the Hecke operators if and only if it satisfies the above Euler product formula.

The connection between modular forms and Dirichlet series is also called Hecke correspondence. With the name Hecke L-series it should be noted that there are also other Hecke L-series, which are formed with generalizations of the Dirichlet characters (size characters according to Hecke) similar to Dirichlet L-series.

literature

• TM Apostol, Modular Functions and Dirichlet Series in Number Theory , Springer Verlag Berlin Heidelberg New York 1990, ISBN 3-540-97127-0
• M. Koecher, A. Krieg, Elliptical functions and modular forms , Springer Verlag Berlin Heidelberg New York 1998, ISBN 3-540-63744-3
• J.-P. Serre: A course in arithmetic , Springer 1973
• LJP Kilford: Modular forms, a classical and computational introduction, Imperial College Press, London 2008

Individual evidence

1. ↑ Hecke "About module functions and the Dirichlet series with Euler's product development" , Math.Annalen, Volume 114, 1937, pp. 1–28
2. ↑ Holomorphic in the upper half-plane
3. ↑ Serre, A course in arithmetic, Springer, p. 98
4. ↑ Atkin, J. Lehner: Hecke operators on , Mathematische Annalen, Volume 185, 1970, pp. 134-160, pdf ${\displaystyle \Gamma _{0}(m)}$
5. ↑ Kilford, Modular Forms, p. 81
6. ↑ Serre, A course in arithmetic, Springer 1973, p. 102