nLab eigenvariety




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


(Emerton09, Definition 2.5)

The p-adic Hecke algebra is the inverse limit

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


(Emerton09, Definition 2.24)

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


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

Last revised on December 2, 2022 at 02:16:22. See the history of this page for a list of all contributions to it.