Independence of $\ell$ for $G$-valued Weil--Deligne representations associated to abelian varieties

Abstract: Let $A$ be an abelian variety over a number field $E$ of dimension $g$ and $\rho_\ell:\mathrm{Gal}(\overline{E}/E)\rightarrow \mathrm{GL}_{2g}(\mathbb{Q}_\ell)$ the Galois representation on the $\ell$-adic Tate module of $A$. For a place $v$ of $E$ not dividing $\ell$, upon fixing an isomorphism $\overline{\mathbb{Q}}_\ell\cong \mathbb{C}$, Grothendieck’s $\ell$-adic monodromy theorem associates to $\rho_\ell$ a $\mathrm{GL}_{2g}(\mathbb{C})$-valued Weil-Deligne representation $\rho_{\ell,v}^{WD}$. Then it is known that the conjugacy class of $\rho_{\ell,v}^{WD}$ is defined over $\mathbb{Q}$ and independent of $\ell.$ When $v$ is a place a good reduction, this is just the result that the characteristic polynomial of Frobenius is defined over $\mathbb{Z}$ and independent of $\ell$.

We consider a refinement of this result. A Theorem of Deligne implies that upon replacing $E$ by a finite extension, the representations $\rho_{\ell,v}^{WD}$ can be refined to a $G(\mathbb{C})$-valued Weil-Deligne representation $\rho^{WD,G}_{\ell,v}$, where $G$ is the Mumford--Tate group of $A$. We prove that for $p>2$ and $v|p$ a place of $E$ where $A$ has semistable reduction, the conjugacy class of $\rho^{WD,G}_{\ell,v}$ is defined over $\mathbb{Q}$ and independent of $\ell$. This is joint work with Mark Kisin.

number theory

Audience: researchers in the topic

London number theory seminar

Series comments: For reminders, join the (very low traffic) mailing list at mailman.ic.ac.uk/mailman/listinfo/london-number-theory-seminar

For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html