헤케 연산자(Hecke operator)

개요

• 모듈라 형식의 공간에 작용하는 연산자
• 헤케 연산자의 고유 벡터가 되는 모듈라 형식의 푸리에 계수는 흥미로운 수론적 성질을 가진다

정의

• \(\mathcal{M}_n\) : 행렬식이 \(n\)인 \(2\times 2\) 정수 계수 행렬들의 집합
• \(f\in M_{k}(\Gamma_1)\), 즉 weight \(k\)인 모듈라 형식,
• 자연수 \(m\)에 대하여, 헤케 연산자 \(T_m\)을 다음과 같이 정의

\[ T_m f(z) = m^{k-1}\sum_{\left(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\right)\in\Gamma\backslash \mathcal{M}_m}(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right) \] 또는, \[ T_m f(z) = m^{k-1}\sum_{a,d>0, ad=m}\frac{1}{d^k}\sum_{b \pmod d} f\left(\frac{az+b}{d}\right)=m^{k-1}\sum_{a,d>0, ad=m}\frac{1}{d^k}\sum_{b=0}^{d-1} f\left(\frac{az+b}{d}\right) \]

• 푸리에 전개가 \(f(z)=\sum_{n=0}^{\infty} c(n) q^n\)로 주어지면, \(T_m f(z)=\sum_{n=0}^{\infty} \gamma_{m}(n) q^n\)이라 할 때, 다음의 관계가 성립

\[ \gamma_{m}(n) = \sum_{r>0, r|(m,n)}r^{k-1}c(\frac{mn}{r^2}) \label{im} \]

• 가령 \(\gamma_{m}(0)=c(0)\sigma_{2k-1}(m)\), \(\gamma_{m}(1)=c(m)\)

성질

• 자연수 \(m,n\) 에 대하여, \(T_{m}T_{n}=T_{n}T_{m}\)
• 서로 소인 자연수 \(m,n\) 에 대하여, \(T_{mn}=T_{m}T_{n} \label{ram1}\)
• 소수 \(p\)와 자연수 \(r\)에 대하여, \(T_{p^{r + 1}} = T_{p}T_{p^r} - p^{k-1}T_{p^{r - 1}} \label{ram2}\)

고유 형식

• weight \(k>0\)인 모듈라 형식 \(f\neq 0\)가 \(\{T_n|n\in \mathbb{N}\}\)의 공통 고유 벡터, 즉 적당한 \(\lambda(m), m\in \mathbb{N}\)에 대하여,

\[ T_mf=\lambda(m)f \] 를 만족할 때, 이를 고유 형식이라 한다

• 고유 형식 \(f\)에 대한 푸리에 전개가 \(f(z)=\sum_{n=0}^{\infty} c(n) q^n\)으로 주어지면,

\[ T_mf(z)=\gamma_m(0)+\gamma_m(1)q+\gamma_m(2)q^2+\cdots=\lambda(m)\left(c(0)+c(1) q^1+c(2)q^2+\cdots\right) \] 이고 \ref{im}로부터 \(\gamma_m(1)=c(m)=\lambda(m)c(1)\)임을 알 수 있다

• 이 때, \(c(1)=1\)이면, 다음이 성립한다

1. 서로 소인 자연수 \(m,n\) 에 대하여, \(c(mn)=c(m)c(n) \)
2. 소수 \(p\)와 자연수 \(r\)에 대하여, \(c(p^{r + 1}) = c(p)c(p^r) - p^{k-1}c(p^{r - 1}) \)

예

• 아이젠슈타인 급수(Eisenstein series)

\[E_{2k}(\tau)=\frac{G_{2k}(\tau)}{2\zeta (2k)}= 1+\frac {2}{\zeta(1-2k)}\left(\sum_{n=1}^{\infty} \sigma_{2k-1}(n)q^{n} \right)\]

• 판별식 함수

\[\Delta(\tau)=\eta(\tau)^{24}= q\prod_{n>0}(1-q^n)^{24}=q-24q+252q^2+\cdots\]

• 자연수의 약수의 합과 라마누잔 타우 함수가 왜 수론적 함수(산술함수, arithmetic function)인지를 이해할 수 있음

메모

• http://mathoverflow.net/questions/153794/using-eichler-selberg-trace-formula-to-compute-dimension-of-modular-forms
• http://mathoverflow.net/questions/101577/are-the-eigenvalues-of-the-hecke-operator-always-real
• http://mathoverflow.net/questions/102395/effect-on-hecke-operator-on-gamman-eisenstein-series

관련된 항목들

• 판별식 (discriminant) 함수와 라마누잔의 타우 함수(tau function)
• 아이젠슈타인 급수(Eisenstein series)
• 수론적 함수(산술함수, arithmetic function)
• 피터슨 내적 (Petersson inner product)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxVjdiaFptbWRHX28/edit

사전 형태의 참고자료

• http://en.wikipedia.org/wiki/Hecke_operator
• http://en.wikipedia.org/wiki/Petersson_inner_product

관련논문

• Kaplan, Nathan, and Ian Petrow. “Traces of Hecke Operators and Refined Weight Enumerators of Reed-Solomon Codes.” arXiv:1506.04440 [cs, Math], June 14, 2015. http://arxiv.org/abs/1506.04440.
• Sakharova, Nina. ‘Convergence of the Zagier Type Series for the Cauchy Kernel’. arXiv:1503.05503 [math], 18 March 2015. http://arxiv.org/abs/1503.05503.
• Holowinsky, Roman, Guillaume Ricotta, and Emmanuel Royer. ‘The Amplification Method in the GL(3) Hecke Algebra’. arXiv:1412.5022 [math], 16 December 2014. http://arxiv.org/abs/1412.5022.
• Popa, Alexandru A. “On the Trace Formula for Hecke Operators on Congruence Subgroups.” arXiv:1408.4998 [math], August 21, 2014. http://arxiv.org/abs/1408.4998.
• Luo, Wenzhi, and Fan Zhou. 2014. “On the Hecke Eigenvalues of Maass Forms.” arXiv:1405.4937 [math], May. http://arxiv.org/abs/1405.4937.
• Takei, Luiz. 2011. On modular forms of weight 2 and representations of PSL(2, Z / pZ). 1103.3066 (March 15). http://arxiv.org/abs/1103.3066.

노트

말뭉치

1. Each Hecke operator has eigenforms when the dimension of is 1, so for , 6, 8, 10, and 14, the eigenforms are the Eisenstein series , , , , and , respectively.
2. Another way to express Hecke operators is by means of double cosets in the modular group.
3. 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 m a 1 and a 1 ≠ 0.
4. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators.
5. Other related mathematical rings are also called "Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious.
6. This is the definition we take then for the Hecke operators of modular forms.
7. Hopefully the above is convincing enough that the Hecke operators as morphisms are fairly natural morphisms to consider when we speak about Hecke operators.
8. In fact, with this algebraic formula, we can de(cid:28)ne Hecke operators on any congruence subgroup .
9. We will consider the case where the Hecke operators act on = 1(N ) (dont care about the weight).
10. This paper focuses on discussing Hecke operators in the theory of modular forms and its relation to Hecke rings which occur in representation theory.
11. We will rst introduce the basic denitions and properties of modular forms and Hecke operators.
12. In this paper, we aim to discuss Hecke operators in the theory of modular forms.
13. We will then introduce Hecke operators and discuss some of their properties.
14. 100 12.5 Hecke operators acting on Jacobians . . . . . . . . . . . . . . . . .
15. In this paper we study the space of period functions of Jacobi forms by means of the Jacobi integral and give an explicit description of the action of Hecke operators on this space.
16. Given the \(q\)-expansion \(f\) of a modular form with character \(\varepsilon\), this function computes the image of \(f\) under the Hecke operator \(T_{n,k}\) of weight \(k\).
17. A Hecke operator is a certain kind of linear transformation on the space of modular forms or cusp forms (see also Modular Forms) of a certain fixed weight .
18. An example of a Hecke operator is the one commonly denoted , for a prime number.
19. Hecke operators are also often defined via their effect on the Fourier expansion of a modular form.
20. The Hecke operators can be defined not only for prime numbers, but for all natural numbers, and any two Hecke operators and commute with each other.
21. In many cases, these coefficients can be recovered from explicit knowledge of the traces of Hecke operators.
22. The original trace formula for Hecke operators was given by Selberg in 1956.
23. This leads to an expression for the trace of a Hecke operator, which is then computed explicitly.
24. Another interesting property of the Hecke operators is that they commute.
25. For any two Hecke operators Tn and Tm dened on Mk, we have the composition formula TmTn = (cid:88) dk1Tmn/d2.
26. We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality be- tween the spaces of holomorphic and weakly holomorphic cusp forms.
27. As an application of this result, we derive some congruences which connect eigenvalues of Hecke operators acting on the space of cusp forms Sk and certain singular moduli.
28. It is my pleasure to thank Ralf Schmidt for generously sharing his insight about Hecke operators with me and for providing the appendix.
29. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms.
30. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.
31. The two equations were proven for (n) by Mordell, using what are now known as the Hecke operators.
32. Following Selberg, we write k(Tn) for the trace of the Hecke operator Tn acting on Sk( (1)).
33. Our starting point is the Selberg trace formula applied to the Hecke operators.
34. Hecke operators 409 Added in proof.
35. The Hecke operators have many applications in various spaces like the space of elliptic modular forms, the space of polynomials and others.
36. We also review some elementary lemmas and theorems both from number theory and ring theory, as well as the de(cid:12)nition of a compact operator and of Hecke operators.
37. Modular forms turn up all over mathematics and physics and, hence, so do the Hecke operators.
38. Algebras of Hecke operators are called Hecke algebras, and the most significant basic fact of the theory is that these are commutative rings.
39. The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Erich Hecke.
40. The idea may be considered to go back to earlier work of Hurwitz, who treated correspondences between modular curves which realise some individual Hecke operator.
41. A third way to express Hecke operators is as double cosets in the modular group.
42. Key words: Hecke operators; vector-valued modular forms; Weil representation.
43. Introduction In Part I of this paper we showed how Hecke operators on L 2 ( S 2 ) may be used to generate very evenly distributed sequences of three-dimensional rotations.
44. In Section 2 of the present paper we give a proof of the inequality (1.2) for a large class of Hecke operators which includes the above example.
45. To begin with we introduce the general Hecke operator: N ( a ) = n LEMMA 2.2.
46. Classically, however, we care about only a few special actions on a few special functions in L2 cusp(G(A), ); namely, how the Hecke operators act on cusp forms of a given weight and level.
47. Let Tn be the Hecke operator on the space of cusp forms Sk(N, !).
48. We want to construct a test function f whose convolution action will mimic the Hecke operator on Sk(N, !) and equal the zero operator on the orthocomplement in L2 cusp(G(A), ).
49. The non-archimedean components will mimic the Hecke operator, while the archimedean component will be cooked up so as to kill the orthocomplement of Sk(!, N ).
50. 2 Hecke operators acts by leftmultiplication on Hn.
51. We see that with the properties of the Hecke operators we can get nice properties for the fourier expansion of these functions.
52. But Hecke showed where these properties come from (the properties of the hecke operator).
53. consisting of eigenfunctions for all Hecke operators, called eigenforms f ( ) = (cid:88) n=1 an(f )qn.
54. In this paper, we dene a graph for each Hecke operator with xed ramication.
55. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms.
56. We develop a structure theory for certain graphs Gx of unramied Hecke operators, which is of a similar vein to Serres theory of quotients of BruhatTits trees.
57. Unramied Hecke operators 3.

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• ID : Q1592959

Spacy 패턴 목록

• [{'LOWER': 'hecke'}, {'LEMMA': 'operator'}]