Let π$\pi$ be a cuspidal automorphic representation of GSp4(boldAboldQ)$\operatorname{GSp}_4(\mathbf {A}_\mathbf {Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If π$\pi$ is not CAP or endoscopic, then we show that its associated ℓ$\ell$‐adic Galois representations are irreducible and crystalline for 100%$100\%$ of primes ℓ$\ell$. If, moreover, π$\pi$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for 100%$100\%$ of primes ℓ$\ell$, the image of its mod ℓ$\ell$ Galois representation contains Sp4(boldFℓ)$\operatorname{Sp}_4(\mathbf {F}_\ell )$.

Let F be a CM field with totally real subfield F + and let π be a C-algebraic cuspidal automorphic representation of the unitary group U(a, b)(A F + ), whose archimedean components are discrete series or non-degenerate limit of discrete series representations. We attach to π a Galois representation Rπ : Gal(F /F + ) → C U(a, b)(Q ℓ ) such that, for any complex conjugation element c, Rπ(c) is as predicted by the Buzzard-Gee conjecture [BG14]. As a corollary, we deduce that the Galois representations attached to certain irregular, C-algebraic essentially conjugate self-dual cuspidal automorphic representations of GLn(AF ) are odd in the sense of Bellaïche-Chenevier [BC11].

For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . Specifically, for any prime $N\equiv 1 \ \pmod p$ , let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point P of order p, and let $B = A_f/\langle P \rangle $ . Then the number of positive integers $d \leq X$ with is $ \gg X/\log X$ , where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$ -type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.