The rationality conjecture for Mumford–Tate-valued Weil–Deligne representations

Let $X$ be a proper, smooth algebraic variety over a number field ${\mathrm{E}}\subset\mathbb C$, and let ${\mathbf{G}}$ be the Mumford–Tate group of the Hodge structure ${\mathrm{H}}^i(X_{\mathbb C},{\mathbb Q})$. For each prime $\ell$, write

$$\rho_{X,\ell}^{{\mathbf{G}}}:\Gamma_{\mathrm{E}}\rightarrow {\mathbf{G}}({\mathbb Q}_\ell)$$

for the associated Galois representation. For a prime $v\nmid\ell$ of ${\mathrm{E}}$, let

$$\rho_{X,\ell,v}^{{\mathrm{WD}},{\mathbf{G}}}:{\mathrm{WD}}_v\rightarrow {\mathbf{G}}({\mathbb Q}_\ell)$$

be the corresponding Weil–Deligne representation, viewed as ${\mathbf{G}}(\mathbb C)$-valued after choosing an isomorphism $\overline{\mathbb Q}_\ell\cong\mathbb C$. Two ${\mathbf{G}}$-valued Weil–Deligne representations are equivalent, written $\rho_1\sim_{\mathbf{G}}\rho_2$, if they are conjugate by an element of ${\mathbf{G}}(\mathbb C)$; such a representation is defined over $\mathbb Q$ if it is equivalent to each of its images under $\operatorname{Aut}(\mathbb C/\mathbb Q)$. The rationality conjecture. There exists a ${\mathbf{G}}$-valued Weil–Deligne representation $\rho_{X,v}^{{\mathrm{WD}},{\mathbf{G}}}$ defined over $\mathbb Q$ such that

$$\rho_{X,v}^{{\mathrm{WD}},{\mathbf{G}}}\sim_{\mathbf{G}}\rho_{X,\ell,v}^{{\mathrm{WD}},{\mathbf{G}}}\quad\text{for all primes }v\nmid\ell.$$

This conjecture expresses the expected rationality, or independence of $\ell$, of the Weil–Deligne representations attached to the cohomology of algebraic varieties. The paper proves the relevant strongly compatible-system statement for semistable abelian varieties, while the general formulation above remains an expectation.