1
$\begingroup$

Let $p$ be a prime and $K$ be a number field. Let $K_\infty$ be a uniform $p$-adic Lie extension of dimension $l$ over $K$ with unique intermediate fields $K_n$ of degree $p^{nl}$ over $K$. We consider an abelian variety $A$ defined over $K$. Are there any precise formulas for $| A(K_n)[p^\infty]|/| A(K_{n-1})[p^\infty]|$?

If $E$ is an elliptic curve with complex multiplication by the ring of integers in an imaginary quadratic field $F$, we can find a number field $F'$ such that $F'(E[p^\infty])/F'$ is a $\mathbb{Z}_p^2$-extension.

Now suppose that $A$ is not an elliptic curve. Are there uniform $p$-adic Lie extension such that $A[p^n]\subset K_n$? And how to construct such an example?

Improve this question
$\endgroup$
1
  • $\begingroup$ The action of the Galois group on the $p$-adic Tate module is a closed subgroup of $GSp_{2g}(\mathbb Z_p)$. There are known results relating such groups to $p$-adic Lie groups (in particular, in Bourbaki) but you have to pass to a finite index subgroup first. In particular, for a typical abelian variety, you will have to adjoin all the $p$-torsion before the uniform $p$-adic Lie extension can be defined. $\endgroup$
    – Will Sawin
    Commented Aug 26, 2021 at 16:40

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .