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Key conjecture: l-adic representations arising from abelian varieties

Establish that any 2g-dimensional Q_ℓ-adic Galois representation ρ of Gal(−/K) that is unramified outside a finite set S and primes above ℓ, has Frobenius traces tr(ρ(Frob_p)) ∈ ℤ for all primes p ∤ ℓ outside S, and is de Rham at the places above ℓ with Hodge–Tate weights {0,1} each of multiplicity g, arises from an abelian variety with complex multiplication as follows: prove the existence of a Galois extension L/K, a CM field E, a degree-one prime λ | ℓ of E, and an abelian variety B/L with an embedding 𝔬_E ↪ End_L(B), good reduction outside S and primes above ℓ, and B ∼_L B^σ for all σ ∈ Gal(L/K), such that the E_λ ≅ Q_ℓ-adic rational Tate module V_λ(B) is isomorphic to ρ as a Gal(−/K)-representation; equivalently, show that for A := Res_K^L(B) one has V_ℓ(A) ≅ ρ^{⊕ [E:ℚ]·[L:K]} as Gal(−/K)-representations.

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Background

The paper’s conditional algorithms rely on a characterization of which ℓ-adic Galois representations come from abelian varieties. The stated conjecture provides a precise criterion: representations with the specified ramification, integrality of Frobenius traces, and de Rham/Hodge–Tate conditions should be realized by the Tate modules of abelian varieties with CM over a suitable Galois extension.

This conjecture is shown to follow from the Hodge, Tate, and Fontaine–Mazur conjectures; thus, resolving it unconditionally would remove the need for those assumptions in the paper’s main algorithmic results.

References

Conjecture \ref{the key conjecture} Let K/Q be a number field. Let \rho \colon Gal(/K) \rightarrow GL_{2g}(Q_\ell) be a Galois representation such that \begin{itemize} \item \rho is unramified outside S and primes above (\ell), \item for every prime p\nmid (\ell) of K not in S, tr(\rho(Frob_p))\in Z, \item and, at each place of K above \ell, the representation \rho is de Rham, with Hodge--Tate weights 0 and 1, each appearing with multiplicity g. \end{itemize}

Then there exists a Galois extension L/K, a CM field E, a degree one prime \lambda\vert (\ell) of E, and an abelian variety B/L admitting \mathfrak{o}E\hookrightarrow End_L(B) with good reduction outside S and primes above \ell such that B\sim_L B\sigma for all \sigma\in Gal(L/K) and moreover the E\lambda\cong Q_\ell-adic rational Tate module V_\lambda(B) := T_\lambda(B)\otimes_{Z_\ell} Q_\ell is isomorphic to \rho as a Gal(/K)-representation: $$V_\lambda(B)\cong \rho.$$

In particular letting A := Res_KL(B) we conclude that V_\ell(A)\cong \rho{\oplus [E : Q]\cdot [L : K]} as Gal(/K)-representations.

— Conditional algorithmic Mordell (2408.11653 - Alpöge et al., 21 Aug 2024) in Conjecture \ref{the key conjecture}, Section “A characterization of Galois representations attached to abelian varieties.”