Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians

Surjectivity of Galois representations in rational families of abelian varieties

“…We show in [LSTX16a, Theorem 1.2] that most members of Y 2g+2,K have monodromy equal to H Y 2g+2,K (which we explicitly compute) over K = Q, and have index-2 monodromy when K = Q. We neither prove nor state this result precisely here, but a complete statement and proof is given in [LSTX16a].…”

“…The above proof of Corollary 1.3 is not constructive. For explicit examples of 1-, 2-, and 3-dimensional PPAVs with maximal adelic Galois representations, see [Gre10, Theorem 1.5] and [Ser72, Sections 5.5.6-8], [LSTX16a], and [Zyw15, Theorem 1.1], respectively.…”

Families of abelian varieties and large Galois images

“…After possibly conjugating ρ E by an element in GL 2g ( Z) we may assume that ρ A (π 1 (U)) is a subgroup of GSp 2g ( Z). Under the "big monodromy" assumption that ρ A (π 1 (U)) is an open subgroup of GSp 2g ( Z), Landesman, Swaminathan, Tao and Xu proved Theorem 1.1 with an optimal C. Earlier, Wallace [Wal14] had proved a variant of this with g = 2; also see Remark 1.3 in [LSTX19]. The case g = 1 had been proved in [Zyw10].…”

“…Many thanks to David Zureick-Brown; this article was originally intended to be part of a joint work with him. Many parts of the original project have been greatly expanded on by his REU students in [LSTX19] and [LSTX17]. In particular, an examination of their papers will hopefully make up for the lack of examples in this article.…”

Surjectivity of the adelic Galois representation associated to a Drinfeld module of rank 3

“…Several mathematicians studied the generalization of adelic surjectivity problem for abelian varieties of dimension 2 over a number field, cf. [AD20], [Hal11], [Zyw15], and [LSTX20]. Currently, it is known that there is a genus 3 hyperelliptic curve over Q whose Jacobian variety satisfies the adelic surjectivity.…”