nLab Hida theory

Idea

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

Definitions

This section follows chapter 8 of DarmonRotgerAWS.

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

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

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

$$\nu(x)=\chi(x)x^{k}$$

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

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

A $\Lambda$-adic modular form of tame level $N$ and tame character $\chi$ is a pair $(\widetilde{\Lambda},\underline{g})$ where $\widetilde{\Lambda}$ is a finite extension of $\Lambda$ and

$$\underline{g}:=\sum \underline{a}_{n}q^{n}$$

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

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

is the q-expansion of a classical normalized ordinary eigenform of weight $k$ and level $N$.

Related concepts

• eigenvariety

• p-adic modular form

References

• 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)