What is "the quotient of the universal ordinary Hecke algebra corresponding to an ordinary $\Lambda$-adic form"?

Question

Let $\Lambda := \Bbb Z_p[[T]]$ be the usual Iwasawa algebra. In Jha and Sujatha - On the Hida deformations of fine Selmer groups on page 181, the authors refer to the quotient $\Bbb H^{\text{ord}}_{\mathcal{F}}$ of the universal ordinary Hecke algebra $\Bbb H^{\text{ord}}_{Np^\infty}$ corresponding to an ordinary $\Lambda$-adic eigenform $\mathcal{F}$. I'm somewhat new to the subject and have been asked to give this paper a go by my masters supervisor, and while I "know" what all of these objects are, Hida's original 1986 paper doesn't explicitly say "the quotient of blah corresponding to blah" and I don't know how to make this connection. Could someone explain this to me like I'm five?

Answer 1

This is very late, but for future reference, but I will post a long answer in case someone who is learning the topic finds this post.

Fix a level $N$ and a prime $p$ such that $p\nmid N$. Just like in the classical case a normalized eigenform $f$ with integral Fourier coefficients (let's say they lie in the ring of integers $\mathcal{O}_F$ of a number ring $F$ or in its $\mathfrak{p}$-adic completion $\mathcal{O}_K$ at a prime $\mathfrak{p}$ above $p$, for now I will denote both of then by $\mathcal{O}$) of weight $k$ and level $\Gamma_0(Np^r)$ defines a morphism of $\mathcal{O}_K$-algebras

$$\mathbb{T}_k(\Gamma_0(Np^r),\mathcal{O})\rightarrow \mathcal{O}$$

By mapping $T_n$ to its eigenvalue on the eigenspace generated by $f$, $a_n(f)$. The kernel is the prime ideal associated to $f$, that is

$$I_f=(\{T_n-a_n(f)\}_n).$$

Notice that the morphism above is what gives the classical duality pairing

$$\mathbb{T}_k(\Gamma_0(Np^r),\mathcal{O})\times S_k(\Gamma_0(Np^r),\mathcal{O}) \rightarrow \mathcal{O}$$

Now this works perfectly fine (up to an extension of scalars) on the ordinary part of universal Hecke algebra $\mathbb{T}(\Gamma_1(N),\mathcal{O}_K)^{ord}$, and it is actually somewhat the definition of Hida family given in Hida's 1986 paper. If this explains enough that's good! Otherwise below there is a more detailed explanation.

What Hida's construction does is essentially considering the universal Hecke algebra on the space $S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}$ of $\mathfrak{p}$-adic modular forms à la Serre. This is the colosure with respect of the $\mathfrak{p},\infty$ norm on $\mathcal{O}_K[[q]]$ (sup norm of $|\cdot|$ on Fourier coefficients) of the image $$\bigcup\limits_{k,r\geq 1} S_k(\Gamma_0(Np^r),\mathcal{O}_k)\rightarrow \mathcal{O}_K[[q]].$$

The word essentially above is needed because he actually lets $k$ vary hile fixing $r$ first and then he let's $r$ vary while fixing $k$, then he shows the closures in $\mathcal{O}_k[[q]]$ of the two spaces obtained this way are the same.

While forming this space one constructs the universal Hecke algebra by taking the limit along system given by the restriction maps given by duality as the dual of the inclusions $$\bigoplus\limits_{k=1}^j S_k(\Gamma_0(Np^{r},\mathcal{O}_K)\rightarrow \bigoplus\limits_{k=1}^{j+1}S_k(\Gamma_0(Np^{r},\mathcal{O}_K),$$ $$S_k(\Gamma_0(Np^{r}),\mathcal{O}_K)\rightarrow S_k(\Gamma_0(Np^{r+1},\mathcal{O}_K).$$ It's important that for the second map the inclusion is $D_{r,1}:f(z)\mapsto f(z)$, not any other choice of degeneracy map so that $U_p$ maps to $U_p$. Since the hecke algebras acts continuously with respect to the norm, these construction agree and form a unique algebra of endomorphism on $S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}$. This is the universal hecke algebra $\mathbb{T}(\Gamma_0(Np^\infty),\mathcal{O}_K)$.

The space $S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}$ has a natural $\Lambda$-action where here $\Lambda$ here is the group algebra $$\mathcal{O}_K[[1+p\mathbb{Z}_p]]:=\lim\limits_{\xleftarrow{r}}\mathcal{O}_K\left[(1+p\mathbb{Z}_p)/1+p^r\mathbb{Z}_p\right].$$ This action is induced by the action of the quotients $$\Gamma_0(Np^r)\backslash \Gamma_0(Np)\cong 1+p\mathbb{Z}_p/1+p^r\mathbb{Z}_p$$ acting trivially on all but finetly many of the components of the limit $$\lim\limits_{\xrightarrow{r\geq 1}} S_k(\Gamma_0(Np^{r},\mathcal{O}_k)$$ Where the map are the degeneracy map of before. Since the action of $\Lambda$ is uniformely continuous it extends to the the space of $\mathfrak{p}$-adic cuspforms and endows it with a $\Lambda$-module structure. By duality we have a $\Lambda$-module structure on $\mathbb{T}(\Gamma_0(Np^\infty),\mathcal{O}_K)$ as well.

As hinted at before the universal Hecke algebra contains the Hecke operator $U_p$. We can then as usual define the ordinary projector as the idempotent $$e^{ord}=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}e^{ord}_t=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}\left(\lim\limits_{n\rightarrow\infty}U^{n!}_p\right).$$ Here $D^{\vee}_{1,r}$ denotes the dual of the degeneracy map defined above (which is a restriction to a specific set of cuspforms which are old at level $Np^{r+1}$) and $e^{ord}_r$ denotes the ordinary projector of level $\Gamma_0(Np^r)$.

We can then define ordinary universal Hecke algerbra and the space of ordinary cuspforms

$$S(\Gamma_1(N),\mathcal{O}_K)^{ord}=e^{ord}\left(S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}\right),$$ $$ \mathbb{T}(\Gamma_0(N),\mathcal{O}_K)^{ord}=\mathbb{T}(\Gamma_0(Np^\infty),\mathcal{O}_K)/(1-e^{ord}).$$

The ordinary universal Hecke algebra is then a finite $\Lambda$-algebra. In particular it is the direct sum of finitely many local rings: $$\mathbb{T}(\Gamma_0(N),\mathcal{O}_K)^{ord}\equiv\bigoplus \limits_{\mathfrak{m}}\mathbb{T}^{ord}_{\mathfrak{m}}.$$

Let's have this sit here for a moment and give a first definition of Hida family of modular form as an irreducible component over $\operatorname{Spec}(\Lambda)$ of $\operatorname{Spec}(\mathbb{T}(\Gamma_0(N),\mathcal{O}_K)^{ord})$. This is a fancy way to say that a Hida family $\mathcal{F}$ is a "minimal" surjective morphism of $\Lambda$-algebras: $\lambda_{\mathcal{F}}:\mathbb{T}(\Gamma_0(N),\mathcal{O}_K)^{ord}\rightarrow\mathcal{R}$ Where $\mathcal{R}$ is some minimal domain of characteristic $0$ containing $\Lambda$. The quotient you are looking for is $\mathcal{R}$ and the ideal is $\ker(\lambda_{\mathcal{F}})$.

Let me now explain what $\mathcal{R}$ actually is. The morphism $\lambda_{\mathcal{F}}$ must factor through a local component $\mathbb{T}^{ord}_{\mathfrak{m}}$ of the ordinary universal Hecke algerbra (Hida says "$\lambda_{\mathcal{F}}$ belongs to $\mathbb{T}^{ord}_{\mathfrak{m}}$"). Then one calls $\mathcal{I}$ the normalization of $\Lambda$ in the extension $$\operatorname{Frac}(\Lambda)\xrightarrow{\Delta} \mathbb{T}^{ord}_{\mathfrak{m}}\otimes_{\Lambda}\operatorname{Frac}(\Lambda).$$

Note that the object on the right is a product of fields (each finite over $\operatorname{Frac}(\Lambda)$ in which we embed $\Lambda$ diagonally. Then $\mathcal{R}$ is the ring $\mathcal{I}$, and it is the ring of Fourier coefficients of $\mathcal{F}$. What happens is that over $\mathcal{I}$ the duality between $\mathbb{T}^{ord}_\mathfrak{m}$ and the set of Hida families passing through $\mathbb{T}^{ord}_{\mathfrak{m}}$ is given by $$(T,\lambda_{\mathcal{F}})\mapsto \lambda_{\mathcal{F}}(T)\in\mathcal{I}.$$

Then one obtain $\mathcal{F}$ as the power series in $\mathcal{I}[[q]]$ as $$\mathcal{F}=\lambda_{\mathcal{F}}(T_n)q^n. $$

We can then describe (over $\mathcal{I}$) the kernel of $\lambda_{\mathcal{F}}$ as $$\mathcal{P}_{\mathcal{F}}=(\{T_n-\lambda_{\mathcal{F}}(T_n)\}).$$