---
date: 2022-04-05 23:42
modification date: Saturday 30th April 2022 14:21:17
title: "Tate module"
aliases: [Tate module, "l-adic Tate module, adelic Tate module", "Tate"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [abelian variety](Unsorted/abelian%20variety.md)

---

# Tate module

Now let's recall the Tate module. The kernel $E[n]$ of the multiply by $n$ map (on $\bar{K}$-points, let's say) consists of the $n$-torsion points of $E$, and is isomorphic to $(\mathbb{Z} / n)^2$. Fixing a prime $\ell$, we have an inverse system
$$
\cdots \longrightarrow E\left[\ell^3\right] \stackrel{\ell}{\longrightarrow} E\left[\ell^2\right] \stackrel{\ell}{\longrightarrow} E[\ell],
$$
whose limit $T_{\ell} E$ we call the $\ell$-adic Tate module of $E$. Each $E\left[\ell^n\right]$ is a (rank-2 free) module over $\mathbb{Z} / \ell^n$, so $T_{\ell} E$ is a (rank 2 free) module over $\lim _{\swarrow} \mathbb{Z} / \ell^n=\mathbb{Z}_{\ell}$.

Furthermore, the $\bar{K}$-points of $E$ get an action by $\operatorname{Gal}(\bar{K} / K)=G_K$, and the group law is rational so $n$-torsion points are sent to $n$-torsion points. This induces an action of $G_K$ on $T_{\ell} E \cong \mathbb{Z}_{\ell}^2$, i.e. a representation
$$
\mathrm{G}_{\mathrm{K}} \rightarrow \mathrm{GL}_2 \mathbb{Z}_{\ell} \subset \mathrm{GL}_2 \mathrm{Q}_{\ell} .
$$
This is continuous because it comes from actions on the finite quotients.
If $E$ has no complex multiplication, then $V_{\ell} E=T_{\ell} E \otimes_{\mathbb{Z}_{\ell}} \mathbf{Q}_{\ell}$ is an irreducible $G_{K^{-}}$ representation for all $\ell$, and $E[\ell]$ is an irreducible $G_K$-representation for almost all $\ell$.


![](attachments/Pasted%20image%2020220503105150.png)

# Motivation

![](attachments/Pasted%20image%2020220430142115.png)

- Studying [moduli stack of abelian varieties](moduli%20stack%20of%20abelian%20varieties) $E$.
- The most important object is the **$\ell\dash$adic  [Tate module](Tate%20module.md) of $E$**, $T_\ell(E)$.
- It is able to detect the ability to lift $E/k$ to the [ring of integers](ring%20of%20integers.md) $O_k$.
- Also captures the [isogeny](isogeny) class of $E$ over a finite field, and the number of points over all finite fields.
- Fails spectacularly when $E/\FF_{p^s}$ is a [supersingular](supersingular) elliptic curve, in which case taking $\ell = p$ yields $T_p E = 0$.
- Leads to considering the [group scheme](group%20scheme.md) $E[\ell^n]$, which is [Unsorted/etale](Unsorted/etale.md) when $\ell \neq p$, but $E[p^n]$ is never étale.
- Leads to replacing $T_p E$ with the directed system $\theset{E[p^n]}_n$.

Define the l-adic Tate module:
$$
T_\ell E := \inverselim_{n} E[\ell^n]
$$
and the adelic Tate module
$$
T_\infty E ;= \inverselim_{n} E[n]
$$

![attachments/Pasted%20image%2020211005011225.png](attachments/Pasted%20image%2020211005011225.png)

![](attachments/Pasted%20image%2020220318204227.png)