Twists of Galois Representations and Projective Automorphisms

Wong, Siman

doi:10.1006/jnth.1998.2307

Cited by 3 publications

(7 citation statements)

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“…Note, for example, that this is a non-trivial condition even if * N 0 =0. It is only under this modified definition of *-adic closeness that the arguments in [9] yield the results as claimed there. However, the statement of [9,Lemma 7] needs to be slightly modified.…”

Section: Introductionsupporting

confidence: 53%

“…Correction to [9]. S. W. would like to take this opportunity to correct a confusing terminology mistake in [9], which is needed in the present paper.…”

Section: Introductionmentioning

confidence: 95%

“…It follows from [9, Cor. 1(b)] (and``Correction to [9]'' below) that if l>5, then the condition (1) implies that the \ E i , l n are equivalent up to twisting by a (ZÂl n ) _ -valued continuous character of G K . For l=3 or l=5, and n>1, the same conclusion holds for the pair of representations G K Ä GL 2 (ZÂl n&1 ) induced from the surjective \ i by reduction modulo l n&1 , thanks to [9, Cor.…”

Section: Introductionmentioning

confidence: 95%

“…1] (see in particular the displayed equation (1) there), \ 1 and \ 2 are defined to be``*-adically close at the supersingular primes'' if there is a positive integer N 0 such that whenever both a i (p) lie in * N 0 , one has for all w N 0 that a 1 (p) # * w if and only if a 2 (p) # * w . This definition is inadequate for the proofs in [9], and is automatically satisfied whenever * N 0 =0 (a case of interest for the present paper)! The definition of *-adic closeness should have been modified to require that if one of the two a i (p) # * N 0 , then for any w N 0 , a 1 (p) # * w if and only if a 2 (p) # * w .…”

Section: Introductionmentioning

confidence: 96%

“…S. W. would like to take this opportunity to correct a confusing terminology mistake in [9], which is needed in the present paper. Let O be a complete local ring with maximal ideal *.…”

Section: Introductionmentioning

confidence: 97%

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Remarks on mod-ln Representations, l=3, 5

Conrad¹,

Wong²

1999

Journal of Number Theory

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Section: Introductionsupporting

confidence: 53%

“…Correction to [9]. S. W. would like to take this opportunity to correct a confusing terminology mistake in [9], which is needed in the present paper.…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 96%

“…S. W. would like to take this opportunity to correct a confusing terminology mistake in [9], which is needed in the present paper. Let O be a complete local ring with maximal ideal *.…”

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Remarks on mod-ln Representations, l=3, 5

Conrad¹,

Wong²

1999

Journal of Number Theory

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On a local-global principle for quadratic twists of abelian varieties

Fité

2022

Math. Ann.

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Let A and $$A'$$ A ′ be abelian varieties defined over a number field k of dimension $$g\ge 1$$ g ≥ 1 . For $$g\le 3$$ g ≤ 3 , we show that the following local-global principle holds: A and $$A'$$ A ′ are quadratic twists of each other if and only if, for almost all primes $$\mathfrak {p}$$ p of k of good reduction for A and $$A'$$ A ′ , the reductions $$A_\mathfrak {p}$$ A p and $$A_\mathfrak {p}'$$ A p ′ are quadratic twists of each other. This result is known when $$g=1$$ g = 1 , in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $$g=4$$ g = 4 .

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Pink-type results for general subgroups of \operatorname{SL}_2(\mathbb{Z}_\ell )^n

Lombardo¹

2017

J. Théor. Nombres Bordeaux

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We study open subgroups G of SL2(Z ℓ ) n in terms of some associated Lie algebras without assuming that G is a pro-ℓ group, thereby extending a theorem of Pink. The result has applications to the study of families of Galois representations.Lemma 2.1. The power series log(1 + t) = n≥1 (−1) n−1 t n n converges for t ∈ ℓZ ℓ (respectively for t ∈ ℓM 2 (Z ℓ )), and establishes a bijection between 1+ℓ 1+v Z ℓ and ℓ 1+v Z ℓ (resp. between B ℓ (1+v) andThe inverse function is given by the power series exp t = n≥0 t n n! , which converges for. Furthermore, for every t ∈ ℓ 1+v Z ℓ we have v ℓ log(1+t) = v ℓ (t), and for every tand n ≥ 2 be an integer. Suppose that A ≡ 0 (mod 4) and2.2 Subgroups of SL 2 (Z ℓ ) n and Z ℓ -Lie algebrasIn this section we consider various properties of closed subgroups of SL 2 (Z ℓ ) n , including their Lie algebras, generating sets, and derived subgroups.Lemma 2.3. Let n be a positive integer and G be a closed subgroup of SL 2 (Z ℓ ) n . The collection of all normal pro-ℓ subgroups of G has a unique maximal element, which we denote by N (G).Likewise, if G is a subgroup of SL 2 (F ℓ ) n , the collection of all normal subgroups of G whose order is a power of ℓ admits a unique maximal element, which we denote again by N (G).Proof. Denote by π : G → SL 2 (F ℓ ) n the canonical projection and let N be a normal, pro-ℓ subgroup of G. Clearly π(N ) is an ℓ-group that is normal in G(ℓ), so it suffices to show thatTo treat the finite case consider first n = 1. Then G is a subgroup of SL 2 (F ℓ ), and it easy to see that the collection of its maximal normal subgroups of order a power of ℓ has a maximal element: it follows from the Dickson classification that this is given by the unique ℓ-Sylow if G is of Borel type, and by the trivial group otherwise. Finally, if G is a subgroup of SL 2 (F ℓ ) n with n > 1, we denote by G i the projection of G on the i-th factor SL 2 (F ℓ ); it is then immediate to check that N (G) = (g 1 , . . . , g n ) ∈ G g i ∈ N (G i ) .Lemma 2.4. Let t be a non-negative integer. Let W ⊆ sl 2 (Z ℓ ) be a Lie subalgebra that does not reduce to zero modulo ℓ t+1 . Suppose that W is stable under conjugation by B ℓ (s) for some non-negative integer s, where s ≥ 2 if ℓ = 2 and s ≥ 1 if ℓ = 3 or 5. Then W contains the open set ℓ t+4s+4v sl 2 (Z ℓ ).Proof. Fix an element w of W that does not vanish modulo ℓ t+1 and write w = µ x x + µ y y + µ h h for some µ x , µ y , µ h ∈ Z ℓ , where

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Twists of Galois Representations and Projective Automorphisms

Cited by 3 publications

References 14 publications

Remarks on mod-ln Representations, l=3, 5

Remarks on mod-ln Representations, l=3, 5

On a local-global principle for quadratic twists of abelian varieties

Pink-type results for general subgroups of \operatorname{SL}_2(\mathbb{Z}_\ell )^n

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