Hecke operator on classical modular forms

经典模形式上的 Hecke 算子

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: full-level Hecke operator

Translation evidence: Literature usage

Meaning in context

On full-level weight-$k$ modular forms, for prime $p$ define $(T_pf)(\tau)=p^{k-1}f(p\tau)+p^{-1}\sum_{b=0}^{p-1}f((\tau+b)/p)$. Set $T_1=I$, $T_{mn}=T_mT_n$ for coprime $m,n$, and $T_{p^{r+1}}=T_pT_{p^r}-p^{k-1}T_{p^{r-1}}$ for $r\ge1$.

在全模群的权 $k$ 模形式空间上，素数指标算子定义为 $(T_pf)(\tau)=p^{k-1}f(p\tau)+p^{-1}\sum_{b=0}^{p-1}f((\tau+b)/p)$；并以 $T_1=I$、互素指标的乘性及 $T_{p^{r+1}}=T_pT_{p^r}-p^{k-1}T_{p^{r-1}}$ 定义其余算子。

Example

For weight $4$, $T_2E_4=9E_4$, since the eigenvalue is $1+2^3$.

Usage and distinctions

This entry fixes full level and the arithmetic normalization; at primes dividing a higher level, the operator has a different formula.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 李文威 — 模形式初步，第六章，作者源文件Proposition prop:T_p and its proof; Definition def:T-general, specialized to N=1 · Checked 2026-10-15
2. 李文威 — 模形式初步，第五章 Hecke 算子通论，作者源文件Section sec:Hecke-full-level, example following eg:full-level-eigenform-1, eigenvalue 1+p^(k-1) of an Eisenstein series · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第五章 Hecke 算子通论，作者源文件第五章标题 Hecke 算子通论；节 sec:modular-form-vs-Hecke-algebra 的正文 · Checked 2026-10-15

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