nLab Hida theory




Hida theory is an approach to p-adic interpolation of modular forms in the ordinary case.


This section follows chapter 8 of DarmonRotgerAWS.

Let Λ\Lambda be the Iwasawa algebra p[[Γ]]\mathbb{Z}_{p}[[\Gamma]] where Γ=(1+p p) ×\Gamma=(1+p\mathbb{Z}_{p})^{\times}. We have that Λ p[[T]]\Lambda\cong \mathbb{Z}_{p}[[T]]. We can further identify Λ\Lambda as the ring of rigid analytic functions on the weight space

𝒳:=Hom cts(Γ, ×).\mathcal{X}:=\Hom_{cts}(\Gamma,\mathbb{C}^{\times}).

A point ν𝒳\nu\in \mathcal{X} is called an arithmetic point if there exists and integer kk (called the weight) and a Dirichlet character χ\chi of p-power order and values in p ×\mathbb{C}_{p}^{\times} such that

ν(x)=χ(x)x k\nu(x)=\chi(x)x^{k}

for all x(1+p p)x\in (1+p\mathbb{Z}_{p}).

More generally a finite extension Λ˜\widetilde{\Lambda} of Λ\Lambda can be identified with the ring of rigid analytic functions on an etale cover 𝒳˜\widetilde{\mathcal{X}} of 𝒳\mathcal{X}.

A Λ\Lambda-adic modular form of tame level NN and tame character χ\chi is a pair (Λ˜,g̲)(\widetilde{\Lambda},\underline{g}) where Λ˜\widetilde{\Lambda} is a finite extension of Λ\Lambda and

g̲:=a̲ nq n\underline{g}:=\sum \underline{a}_{n}q^{n}

where a̲ nΛ˜\underline{a}_{n}\in \widetilde{\Lambda}, such that for almost all νX˜\nu\in\widetilde{X} of weight k2k\in\mathbb{Z}\geq 2, the power series

g̲:=a̲ n(ν)q n\underline{g}:=\sum \underline{a}_{n}(\nu)q^{n}

is the q-expansion of a classical normalized ordinary eigenform of weight kk and level NN.


  • Chris Williams, An Introduction to Hida Theory (pdf)

  • Cameron Franc, Hida Theory (pdf)

A discussion may also be found in chapter 8 of

  • Henri Darmon and Victor Rotger, Algebraic Cycles and Stark-Heegner Points, (pdf)

Last revised on December 2, 2022 at 17:00:39. See the history of this page for a list of all contributions to it.