Principally polarized abelian surfaces with surjective Galois representations on <i>l</i>
-torsion

Wallace, E. W.

doi:10.1112/jlms/jdu033

Cited by 4 publications

(7 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let δ Q be the index of the closure of the commutator subgroup of H A in H A ∩ Sp 2g ( Z), and let δ K = 1 for K = Q. Then [H A : H Au ] ≥ δ K for all u ∈ U(K), and we have the following asymptotic statements: where the implied constants depend only on A → U and n. [Wal14,p. 468] how to correct some of the errors in Kawamura's proof, the modified argument still appears to be mistaken; see [Lom16,p.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…Then, in Section 4.3, we introduce some of the notation and standing assumptions employed in the proof. In particular, since our family has big geometric monodromy, by Proposition 4.1, we are able to define the constant C in point (b) of Section 4.3, which will later be needed to apply the results of [Wal14] (see Section 4.6.1).…”

Section: Background On Galois Representations Of Ppavsmentioning

confidence: 99%

“…Remark 1.2. In [Wal14], Wallace studies a variant of Theorem 1.1 in the 2-dimensional case. Unfortunately, his argument relies upon a mistaken Masser-Wüstholz-type result of Kawamura, [Kaw03, Main Theorem 2].…”

Section: Question How Large Can the Image Of The Adelic Galois Repres...mentioning

confidence: 99%

“…Unfortunately, his argument relies upon a mistaken Masser-Wüstholz-type result of Kawamura, [Kaw03, Main Theorem 2]. Although Wallace describes in [Wal14,p. 468] how to correct some of the errors in Kawamura's proof, the modified argument still appears to be mistaken; see [Lom16,p.…”

Section: Question How Large Can the Image Of The Adelic Galois Repres...mentioning

confidence: 99%

“…In this section, we set up notation for extending a given rational family of PPAVs over a number field K to a family over the number ring O K . This construction will become particularly important in Section 4.6, where we apply the results of [Wal14].…”

Section: Background On Galois Representations Of Ppavsmentioning

confidence: 99%

See 4 more Smart Citations

Surjectivity of Galois representations in rational families of abelian varieties

et al. 2019

View full text Add to dashboard Cite

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-1 subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension g ≥ 3, there are infinitely many abelian varieties over Q with adelic Galois representation having image equal to all of GSp 2g ( Z).

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Background On Galois Representations Of Ppavsmentioning

confidence: 99%

Section: Question How Large Can the Image Of The Adelic Galois Repres...mentioning

confidence: 99%

Section: Question How Large Can the Image Of The Adelic Galois Repres...mentioning

confidence: 99%

Section: Background On Galois Representations Of Ppavsmentioning

confidence: 99%

See 3 more Smart Citations

Surjectivity of Galois representations in rational families of abelian varieties

et al. 2019

View full text Add to dashboard Cite

show abstract

Lifting subgroups of symplectic groups over "Equation missing"

et al. 2017

View full text Add to dashboard Cite

For a positive integer g, let Sp 2g (R) denote the group of 2g × 2g symplectic matrices over a ring R. Assume g ≥ 2. For a prime number , we give a self-contained proof that any closed subgroup of Sp 2g (Z ) which surjects onto Sp 2g (Z/ Z) must in fact equal all of Sp 2g (Z ). The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.

show abstract

Families of Abelian Varieties and Large Galois Images

Zywina

2022

International Mathematics Research Notices

View full text Add to dashboard Cite

Associated with an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho _A\colon \operatorname {Gal}({\overline {K}}/K)\to \operatorname {GL}_{2g}(\widehat {{\mathbb {Z}}})$. The representation $\rho _A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a non-empty open subvariety $U\subseteq {\mathbb {P}}^n_K$. We show that away from a set of density $0$, the image of $\rho _A$ will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results that assumed that the family of abelian varieties has “big monodromy”. We also give a version for families of abelian varieties with a more general base.

show abstract

Principally polarized abelian surfaces with surjective Galois representations on l -torsion

Cited by 4 publications

References 30 publications

Surjectivity of Galois representations in rational families of abelian varieties

Surjectivity of Galois representations in rational families of abelian varieties

Lifting subgroups of symplectic groups over "Equation missing"

Families of Abelian Varieties and Large Galois Images

Contact Info

Product

Resources

About