[Paper Review] Hecke operators in half-integral weight

Soma Purkait|arXiv (Cornell University)|Aug 21, 2012

Advanced Algebra and Geometry4 references3 citations

TL;DR

This paper establishes analogues of Sturm's bounds and Hecke operator recursion relations for modular forms of half-integral weight, proving that the Hecke algebra on the orthogonal complement of theta-series is generated by $T_{p^2}$ for primes $p \leq R$, where $R$ is a bound derived from the level and weight. It further provides explicit formulas for how Hecke operators $T_{p^{2\ell}}$ commute with the Shimura correspondence, extending known results to all primes, including those dividing the level.

ABSTRACT

In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^\ell}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators $T_{p^{\ell}}$ commute with the Shimura correspondence.

Motivation & Objective

• To extend Sturm's bounds on Hecke algebra generators from integral to half-integral weight modular forms.
• To establish recursion formulas for Hecke operators $T_{p^{2\ell}}$ in the half-integral weight setting, analogous to those in integral weight.
• To determine the precise commutation relations between Hecke operators $T_{p^{2\ell}}$ and the Shimura correspondence for all primes $p$, including those dividing the level $N$.
• To prove that the restriction of the Hecke algebra to the non-theta subspace $S_{k/2}^\perp(\chi)$ is generated by $T_{p^2}$ for $p \leq R$, with $R$ explicitly computed.

Proposed method

• Derives recursion relations for $T_{p^{2\ell}}$ in half-integral weight: $T_{p^{2\ell+2}} = T_{p^2}T_{p^{2\ell}} - \chi(p^2)p^{k-2}T_{p^{2\ell-2}}$ when $p \nmid N$, and $T_{p^{2\ell}} = (T_{p^2})^\ell$ when $p \mid N$.
• Uses the Shimura correspondence $\operatorname{Sh}_t: S_{k/2}(N,\chi) \to M_{k-1}(N', \chi^2)$ to relate half-integral and integral weight Hecke algebras.
• Applies the theory of $q$-expansions and Hecke operator actions via $q$-series to analyze operator behavior.
• Employs the formula $T_p(f) = \sum (a_{pn} + \chi(p)p^{k-1}a_{n/p})q^n$ to express Hecke actions in terms of Fourier coefficients.
• Uses Sturm's bound (Theorem 8) and group index formulas for $[\mathrm{SL}_2(\mathbb{Z}) : \Gamma_1(N')]$ and $[\mathrm{SL}_2(\mathbb{Z}) : \Gamma_0(N')]$ to derive the bound $R$.
• Applies module-theoretic arguments over $\mathbb{Z}[\zeta_{\varphi(N)}]$ to show that $T_{i^2}$ for $i \leq R$ generate $\mathbb{T}_{k/2}^\perp$.

Experimental results

Research questions

• RQ1 Can Sturm-type bounds on Hecke algebra generators be extended from integral to half-integral weight modular forms?
• RQ2 What recursion relations govern the Hecke operators $T_{p^{2\ell}}$ in the half-integral weight setting, especially when $p \mid N$?
• RQ3 How do the Hecke operators $T_{p^{2\ell}}$ commute with the Shimura correspondence for all primes $p$, including those dividing the level $N$?
• RQ4 What is the minimal set of Hecke operators $T_{p^2}$ that generate the Hecke algebra on the non-theta subspace $S_{k/2}^\perp(N,\chi)$?
• RQ5 How does the bound $R$ for generator finiteness depend on the level $N$, weight $k$, and character $\chi$?

Key findings

• For $p \mid N$, the Hecke operator satisfies $T_{p^{2\ell}} = (T_{p^2})^\ell$ in $\mathbb{T}_{k/2}$.
• For $p \nmid N$, the recursion $T_{p^{2\ell+2}} = T_{p^2}T_{p^{2\ell}} - \chi(p^2)p^{k-2}T_{p^{2\ell-2}}$ holds in $\mathbb{T}_{k/2}$.
• The Shimura correspondence satisfies $\operatorname{Sh}_t(T_{p^2}f) = T_p(\operatorname{Sh}_t(f))$ for all primes $p$, even when $p \mid tN$, extending a known result.
• For $\ell \geq 2$, $\operatorname{Sh}_t(T_{p^{2\ell}}f) = T_{p^\ell}(\operatorname{Sh}_t(f))$ if $p \mid N$, and $\operatorname{Sh}_t(T_{p^{2\ell}}f) = (T_{p^\ell} - \chi(p^2)p^{k-3}T_{p^{\ell-2}})(\operatorname{Sh}_t(f))$ if $p \nmid N$.
• The Hecke algebra $\mathbb{T}_{k/2}^\perp$ is generated as a $\mathbb{Z}[\zeta_{\varphi(N)}]$-module by $T_{i^2}$ with $i \leq R$, where $R = \frac{(k-1)m}{12} - \frac{m-1}{N'}$, with $m = N'^2 \prod_{p \mid N'} (1 - p^{-2})$.
• When $\chi$ is quadratic, the same result holds with $m = N' \prod_{p \mid N'} (1 + p^{-1})$, showing a refined bound for this case.