On congruences of Galois representations of number fields. (English) Zbl 1310.11070
This paper considers \(\ell\)-adic representations \(V\) of the absolute Galois group of an algebraic number field \(K\). For a place \(v\) of \(K\), one can consider a decomposition group of \(K\) above \(v\), and the corresponding \(\ell\)-adic representation \(V_v\). The main result of the paper is as follows: Given a place \(v\), and two \(\ell\)-adic representations \(V\) and \(V'\) satisfying certain assumptions (which are satisfied when \(V\) and \(V'\) “come from geometry,” i.e., are representations in the \(\ell\)-adic cohomology of a smooth proper variety), then for large enough \(\ell\) and any place \(u\) of \(K\) over \(\ell\), having an isomorphism between the semisimplifications of \(V_u\) and \(V'_u\) modulo \(\ell\), as well as an isomorphism between the semisimplifications of \(V_v\) and \(V'_v\) modulo \(\ell\), is sufficient to conclude that there is an isomorphism between the semisimplifications of \(V_v\) and \(V_v'\). Several variants of this statement are proven as well, involving weaker assumptions and weaker conclusions, as well as generalizing to the case where the coefficients lie in a finite extension of \(\mathbb{Q}_{\ell}\).
The proofs of these statements are relatively short, but rely on several deep results in \(p\)-adic Hodge theory.
The main result is applied to a generalization of the conjecture of Rasmussen and Tamagawa, which states that for large enough \(\ell\), there does not exist an abelian variety \(A\) of a given dimension over a given number field with good reduction outside \(\ell\) such that the Galois representation of the decomposition group of \(v\) on \(A[\ell]\) is Borel with diagonal components powers of the mod \(\ell\) cyclotomic character. If \(u\) is a place of \(K\) and \(E_{\lambda}/\mathbb{Q}_{\ell}\) is a finite extension, then the analogous statement is proven for the representation on \(H^r(X, E_{\lambda})\), where \(X\) is a smooth proper \(K\)-variety having semistable reduction at \(u\) and potentially good reduction at \(v\).
A further application is given involving congruences of coefficients of cusp forms.
The proofs of these statements are relatively short, but rely on several deep results in \(p\)-adic Hodge theory.
The main result is applied to a generalization of the conjecture of Rasmussen and Tamagawa, which states that for large enough \(\ell\), there does not exist an abelian variety \(A\) of a given dimension over a given number field with good reduction outside \(\ell\) such that the Galois representation of the decomposition group of \(v\) on \(A[\ell]\) is Borel with diagonal components powers of the mod \(\ell\) cyclotomic character. If \(u\) is a place of \(K\) and \(E_{\lambda}/\mathbb{Q}_{\ell}\) is a finite extension, then the analogous statement is proven for the representation on \(H^r(X, E_{\lambda})\), where \(X\) is a smooth proper \(K\)-variety having semistable reduction at \(u\) and potentially good reduction at \(v\).
A further application is given involving congruences of coefficients of cusp forms.
Reviewer: Andrew Obus (Charlottesville)
Keywords:
\(\ell\)-adic Galois representation; congruence; semisimplification; Hodge-Tate weights; Weil weightsReferences:
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