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

• Tags:
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • abelian variety

Tate module

Motivation

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

Define the l-adic Tate module: \begin{align*} T_\ell E := \varprojlim_{n} E[\ell^n] \end{align*} and the adelic Tate module \begin{align*} T_\infty E ;= \varprojlim_{n} E[n] \end{align*}