Principally polarized abelian surfaces with surjective Galois representations on \(l\)-torsion (Q2922847)

Let \(E\) be an elliptic curve over \(\mathbb Q\), without complex multiplication, and \(l\) be a prime number. Then the absolute Galois group acts on the \(l\)-torsion sub-group \(E[l]\) as a subgroup of \(\mathrm{Aut}(E[l])\simeq \mathrm{GL}_2(\mathbb{F}_l)\). It is expected that this subgroup is in fact the full group \(\mathrm{GL}_2(\mathbb{F}_l)\) and this is the case for all but finitely many primes \(l\). Also, a theorem of \textit{W. Duke} [C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 8, 813--818 (1997; Zbl 1002.11049)] states that the density of elliptic curves defined over \(\mathbb Q\) admitting at least one `exceptional' prime \(l\), i.e. a prime \(l\) for which the mentioned representation is not surjective, is zero. A generalization over number fields was provided by \textit{D. Zywina} [Bull. Lond. Math. Soc. 42, No. 5, 811--826 (2010; Zbl 1221.11136)].NEWLINENEWLINEIn the paper under review, the author considers the natural analogue for abelian surfaces: namely, given a family of abelian surfaces over an algebraic variety, it is proved that for almost all of them the Galois action on their points of \(l\)-torsion, for every prime \(l\), is given by the full group \(\mathrm{GSP}_{2g}({\mathbb F}_l)\) of symplectic similitudes.NEWLINENEWLINEMore precisely: let \(K\) be a number field, \(V\) a smooth geometrically irreducible affine variety of dimension \(3\), \(\mathcal{A}\to V\) a principally polarized abelian scheme over \(V\) of relative dimension \(2\), inducing a dominant map to the moduli space of principally polarized abelian surfaces. Let us fix a height function on \(V(\bar{K})\) and consider, for each \(x\geq 1\), the set \(E_K(x)\) of points \(t\in V(K)\) of height \(\leq x\) such that the abelian surface \(A_t\) has the following property: for at least one prime \(l\) the Galois action on \({\mathcal A}_t[l]\) is not surjective on the group \(\mathrm{GSP}_{2g}({\mathbb F}_l)\). Finally, denote by \(B_K(x)\) the set of all the \(K\)-rational points of height \(\leq x\) on \(V\).NEWLINENEWLINETheorem 1.1 then states that the ratio \(|E_K(x)|/|B_K(x)|\) tends to zero, and actually is bounded as NEWLINE\[NEWLINE {|E_K(x)|\over |B_K(x)|}\ll_{K,V} {(\log x)^{17}\over x^{1/2}}. NEWLINE\]NEWLINE Reviewer's remark: The condition that the abelian varieties \({\mathcal A}_t\) really form a \(3\)-dimensional family, for instance they are not constant in \(t\in V\), does not appear explicitly in Theorem 1.1, but is clearly needed. It is stated by the author in \S 4.