Related papers: Adelic openness without the Mumford-Tate conjectur…

Let $Y$ be an abelian variety over a subfield $k \subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford-Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre…

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the…

In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying…

Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the…

Let $X$ be a complete smooth variety defined over number field $K$ and $i$ an integer. The absolute Galois group of $K$ acts on the $i$th $l$-adic etale cohomology of $X$ for all $l$, producing a system of $l$-adic representations…

We show that the Andr\'{e} motive of a hyper-K\"{a}hler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent…

Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\mathrm{Gal}(K)$ on the \'etale…

Let $K$ be a number field and $E_1, \ldots, E_n$ be elliptic curves over $K$, pairwise non-isogenous over $\overline{K}$ and without complex multiplication over $\overline{K}$. We study the image of the adelic representation of the absolute…

Let E be an elliptic curve over the rationals without complex multiplication. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation rho_E:Gal(Qbar/Q)…

For a non-CM elliptic curve $E$ defined over a number field $K$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{K}/K)\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A…

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…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image…

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when…

For a non-CM elliptic curve $E$ defined over $\mathbb{Q}$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to…

Let $k$ be a finitely generated field of characteristic $p > 0$ and $\ell$ a prime. Let $X$ be a smooth, separated, geometrically connected curve of finite type over $k$ and $\rho: \pi_1(X)\rightarrow GL_r(\mathbb Z_{\ell})$ a continuous…

Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $\rho^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…