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Bounding the number of generators

Let $ \mu(N)=N\prod_{p\vert N}(1+\frac{1}{p})$ be the index of $ \Gamma_0(N)$ in $ \SL_2(\mathbb{Z})$.

Theorem 2.1   Let $ \lambda$ be a prime ideal in the ring of integers $ \O $ of some number field. Suppose $ f\in M_k(N;\O )$ is such that $ a_n(f)\equiv 0\pmod{\lambda}$ for $ n\leq \frac{k}{12}\mu(N)$. Then $ f\equiv 0\pmod{\lambda}$.

Proof. Theorem 1, [S]. $ \qedsymbol$

Denote by $ \lceil{}x\rceil$ the smallest integer $ \geq x$.

Proposition 2.2   Suppose $ f\in M_k(N)$ and

$\displaystyle a_n(f)=0$   for$\displaystyle \quad
n\leq r=\left\lceil\frac{k}{12}\mu(N)\right\rceil.$

Then $ f=0$.

Proof. We must show that the composite map

$\displaystyle M_k(N)\hookrightarrow\mathbb{C}[[q]]\rightarrow \mathbb{C}[[q]]/(q^{r+1})$

is injective. Because $ \mathbb{C}$ is a flat $ \mathbb{Z}$-module, it suffices to show that the map $ \Phi:M_k(N;\mathbb{Z})\rightarrow \mathbb{Z}[[q]]/(q^{r+1})$ is injective. Suppose $ \Phi(f)=0$, and let $ p$ be a prime number. Then $ a_n(f)=0$ for $ n\leq r$, hence plainly $ a_n(f)\equiv 0\pmod{p}$ for any such $ n$. By Theorem 2.1, it follows that $ f\equiv 0\pmod{p}$. Repeating this argument shows that the coefficients of $ f$ are divisible by all primes $ p$, i.e., they are 0. $ \qedsymbol$

Theorem 2.3   The Hecke algebra is generated as a $ \mathbb{Z}$-module by $ T_1,\ldots,T_r$ where $ r=\lceil \frac{k}{12}\mu(N)\rceil $. Thus displaymathT=Z[T_1,T_2,T_3,...] = Z{T_1,...,T_r}.

Proof. Let $ A$ be the submodule of $ \mathbb{T}$ generated by $ T_1,T_2,\ldots,T_r$. Consider the exact sequence of additive abelian groups

$\displaystyle 0\rightarrow A \xrightarrow{i} \mathbb{T}\rightarrow \mathbb{T}/A \rightarrow 0.$

Let $ p$ be a prime and tensor with $ \mathbb{F}_p$ to obtain

$\displaystyle A\otimes \mathbb{F}_p\xrightarrow{\overline{i}} \mathbb{T}\otimes \mathbb{F}_p \rightarrow (\mathbb{T}/A)\otimes \mathbb{F}_p\rightarrow 0$

(tensor product is right exact). Put $ R=\mathbb{F}_p$ in Proposition 1.2, and suppose $ f\in M_k(N,\mathbb{F}_p)$ pairs to 0 with each of $ T_1,\ldots,T_r$. Then by Proposition 1.1, $ a_m(f)=a_1(T_m f)=0$ in $ \mathbb{F}_p$ for each $ m$, $ 1\leq m\leq r$. By Theorem 2.1 it follows that $ f=0$. Thus the pairing, when restricted to the image of $ A\otimes \mathbb{F}_p$ in $ \mathbb{T}\otimes \mathbb{F}_p$, is also perfect and so

$\displaystyle \dim_{\mathbb{F}_p} \overline{i}(A\otimes \mathbb{F}_p)
= \dim_{...
...{F}_p} M_k(N,\mathbb{F}_p)= \dim_{\mathbb{F}_p} \mathbb{T}\otimes \mathbb{F}_p.$

We see that $ (\mathbb{T}/A) \otimes \mathbb{F}_p = 0$; repeating the argument for all $ p$ shows that the finitely generated abelian group $ \mathbb{T}/A$ must be trivial. $ \qedsymbol$

Let tex2html_wrap_inline$S_k^(N)$ be the new subspace of tex2html_wrap_inline$S_k(N)$. It is the orthogonal complement, with respect to the Peterson pairing (VII, §5, [L]), of the subspace spanned by the images of tex2html_wrap_inline$S_k(M)$ for tex2html_wrap_inline$M|N$ under the natural inclusion maps. Let tex2html_wrap_inline$T^$ be the image of the Hecke algebra in the ring of endomorphisms of tex2html_wrap_inline$S_k^(N)$. By (VIII, §3, [L]), tex2html_wrap_inline$S_k^(N)$is a direct sum of distinct one dimensional eigenspaces. We call tex2html_wrap_inline$f&isin#in;S_k^(N)$ a newform if it is an eigenform for all Hecke operators tex2html_wrap_inline$T_p$ and if it is normalized so that tex2html_wrap_inline$a_1(f)=1$.

theorem_type[proposition][theorem][][plain][][] If tex2html_wrap_inline$f$ is a newform level tex2html_wrap_inline$N$ and tex2html_wrap_inline$p|N$, then displaymatha_p(f) = cases±p^k/2 -1& if $p||N$
0 & if $p^2|N$.proof See the end of §6 in [DI].

Fix a square free positive integer tex2html_wrap_inline$N$. Let tex2html_wrap_inline${p_1,p_2,...p_s}$ be a subset (possibly empty, in which case tex2html_wrap_inline$s=0$) of the prime divisors of tex2html_wrap_inline$N$and set displaymathr:=[k&mu#mu;(N)(12·2^s)].

theorem_type[theorem][theorem][section][plain][][] Let tex2html_wrap_inline$&lambda#lambda;$ be a prime ideal in the ring of integers tex2html_wrap_inline$O$ of some number field. Suppose tex2html_wrap_inline$f$ and tex2html_wrap_inline$g$ are newforms in tex2html_wrap_inline$S_k^(N;O)$. Assume enumerate tex2html_wrap_inline$a_n(f-g) &equiv#equiv;0 &lambda#lambda;$ for tex2html_wrap_inline$n&le#leq;r$ and tex2html_wrap_inline$a_p_i(f) = b_p_i(g)$ for each tex2html_wrap_inline$i=1,...s$. Then tex2html_wrap_inline$f&equiv#equiv;g&lambda#lambda;$. proof Theorem 2 of [S]. Note that by Proposition [*] tex2html_wrap_inline$a_p(f) = ±b_p(f)$.

I wonder: is displaymathT^=Z{ T_1,..., T_r, T_p_1,..., T_p_s}?I DON'T see that it does because theorem 3.5 says that, essentially, the first vectors of first tex2html_wrap_inline$r$ entries of a basis of eigenforms are all different. But, there's no reason I can see that they have to be linearly independent.


next up previous
Next: Bibliography Up: generating_hecke Previous: Modular forms and Hecke
William A Stein 2002-02-01