Let $X$ be an abelian surface over a finite extension of $\mathbb{Q}_p$. When does $X$ have distinct Hodge--Tate weights (in étale cohomology)?
$\begingroup$
$\endgroup$
4
-
4$\begingroup$ By Hodge-Tate comparison for abelian varieties (proved by Tate), Hodge-Tate weights are the same as Hodge numbers, so for $X$ an abelian variety of dim $g$ they are $\binom{g}{i} \binom{g}{n-i}$ in degree $n$. These are never all $\leq 1$ (I guess this is what you mean by distinct Hodge-Tate weights) unless $g=1$. $\endgroup$– Piotr AchingerCommented May 15, 2019 at 17:10
-
2$\begingroup$ @PiotrAchinger do you mean that the multiplicity of the Hodge--Tate weight $i$ is the product of binomial coefficients? or am I misunderstanding the terminology? $\endgroup$– user138661Commented May 15, 2019 at 17:32
-
4$\begingroup$ I'm claiming that $\binom{g}{i} \binom{g}{n-i}$ is the multiplicity of $\mathbf{C}_p(-i)$ in the Hodge-Tate decomposition of $H^n(X_{\bar K}, \mathbf{Q}_p)$. This follows from the fact that $H^n = \bigwedge^n H^1$ as Galois representations, and that $H^1$ has Hodge-Tate weights $0$ and $1$, both with multiplicity $g$. $\endgroup$– Piotr AchingerCommented May 15, 2019 at 20:21
-
1$\begingroup$ @PiotrAchinger but then the original comment was slightly sloppily worded: "Hodge-Tate weights...they are...". Do you agree? $\endgroup$– user138661Commented May 16, 2019 at 7:06
Add a comment
|