nLab eigenvariety

Idea

An eigenvariety is a rigid analytic space that parametrizes p-adic families of systems of Hecke eigenvalues. Historically the first example is the eigencurve as constructed by Coleman and Mazur.

Construction of the Eigencurve

Let $T_{\ell}$ and $S_{\ell}$ be the Hecke operators corresponding to the double cosets $\GL_{\mathbb{Z}_{\ell}}\begin{pmatrix}\ell & 0 \\ 0 & 1\end{pmatrix}\GL_{\mathbb{Z}_{\ell}}$ and $\GL_{\mathbb{Z}_{\ell}}\begin{pmatrix}\ell & 0 \\ 0 & \ell\end{pmatrix}\GL_{\mathbb{Z}_{\ell}}$ respectively. Let $\mathcal{M}_{k}(N)$ be the space of modular forms of weight $k$ and level $\Gamma_{1}(N)$. Let $\mathbb{T}_{\leq k}^{(p)}$ be the ring of endomorphisms of $\bigoplus_{i=1}^{k}\mathcal{M}_{i}(N)$ generated by the Hecke operators $T_{\ell}$ and $S_{\ell}$ as $\ell$ ranges over all primes not dividing $N p$.

Definition

(Emerton09, Definition 2.5)

The p-adic Hecke algebra is the inverse limit

$$\mathbb{T}:=\underset {\underset{k}{\leftarrow}} {\lim} \;\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{T}_{\leq k}^{(p)}$$

Definition

(Emerton09, Definition 2.24)

Let $\mathcal{X}$ be the set of $\overline{\mathbb{Q}}_{p}$-valued points in $\Spec\mathbb{T}\times\mathbb{G}_{m}$ consisting of pairs $(\xi,\alpha)$ where $\xi:\mathbb{T}\to \overline{\mathbb{Z}}_{p}$ is classical, corresponding to some system of Hecke eigenvalues $\mathbb{T}_{k}\to \overline{\mathbb{Z}}_{p}$ and $\alpha$ is a root of the $p$-th Hecke polynomial. The eigencurve is the rigid analytic Zariski closure of $\mathcal{X}$ in $(\Spec\mathbb{T}\times\mathbb{G}_{m})^{\an}$.

Related concepts

• p-adic modular form

• Hida theory

References

• Matthew Emerton, p-adic Families of Modular Forms, 2009 (pdf)