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Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, defined over $K$, and having good reduction at $v$. For each such $A$ there is an attached crystalline representation $\rho_A$ of $\text{Gal}(\overline{K_v}/K_v)$.

What are the properties of the set $S_F = \{\rho_A : A \in F(K,g,v)\}$?

For example, when $g = 1, K = \mathbb{Q}, p \geq 5$ then $S_F$ is naturally partitioned into representations that come from elliptic curves with super-singular good reduction at $p$ and those coming from ordinary good reduction. These two subsets are distinguished by their Newton polygons.

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  • $\begingroup$ Why restrict to $K=\mathbb{Q}$ and $p\geq5$ in the $g=1$ case? Isn't this true in general when $g=1$? In general such a representation $\rho_A$ will have a crystalline Frobenius, and if you choose to partition $S_F$ up according to the slopes of this semilinear map then you will have an analogy of your decomposition in the $g=1$ case. $\endgroup$
    – Kevin Buzzard
    Commented Dec 7, 2018 at 21:16
  • $\begingroup$ @KevinBuzzard the most direct analogue in the higher dimensional case is if one sorts $S_F$ by the possible characteristic polynomials of Frobenius, which are necessarily Weil polynomials of degree $2g$. There are only finitely many of these (alternatively, the reductions mod $\mathbb{F}_v$ of the $A$'s determine these polynomials, and there are only finitely many abelian varieties defined over $\mathbb{F}_v$). I was hoping for a slightly more refined partition than this $\endgroup$
    – Stanley Yao Xiao
    Commented Dec 7, 2018 at 23:53

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