Related papers: Comparison of different Tate conjectures

The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having…

In this note we prove new cases of the Mumford-Tate conjecture by extending a theorem of Richard Pink for abelian varieties without nontrivial endomorphisms and with bad semistable reduction. We use quadratic pairs introduced by…

In this paper, we establish two main results concerning the Mumford-Tate conjecture for hyper-K\"ahler varieties. First, we prove the conjecture for the semisimplified $\ell$-adic Galois representations attached to hyper-K\"ahler varieties…

Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally…

We develop the theory and algorithms necessary to be able to verify the strong Birch--Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over $\mathbf{Q}$. We apply our methods to all 28 Atkin--Lehner quotients of…

Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank…

Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of…

Let $A$ be an abelian variety defined over a number field $K$. For a finite extension $L/K$, the cardinality of the group $A(L)_{\operatorname{tors}}$ of torsion points in $A(L)$ can be bounded in terms of the degree $[L:K]$. We study the…

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…

Suppose $T$ is totally transcendental and every minimal non-locally-modular type is nonorthogonal to a nonisolated minimal type over the empty set. It is shown that a finite rank type $p=tp(a/A)$ is isolated if and only if $a$ is…

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we…

Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the…

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasiprojective variety. Let $\mathcal{E}$ be a semisimple overconvergent $F$-isocrystal on $X$. Suppose that irreducible summands $\mathcal{E}_i$ of $\mathcal E$ have rank 2,…

Let A be a complex abelian variety and G its Mumford--Tate group. Supposing that the simple abelian subvarieties of A are pairwise non-isogenous, we find a lower bound for the rank of G, which is a little less than log_2 dim A. If we…

Let A be an abelian variety over a number field k. We show that weak approximation holds in the Weil-Ch\^atelet group of A/k but that it may fail when one restricts to the n-torsion subgroup. This failure is however relatively mild; we show…

Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if…

Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. When $A$ is isogenous to a product of simple abelian…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

Let A be an abelian variety over a local field K of mixed characteristic and with algebraically closed residue field. We provide a geometric construction (via the relative Picard functor) of the Shafarevich duality between the group of…

Given an abelian variety J and an abelian subvariety A of J over a number field K, we study the visible elements of the Shafarevich-Tate group of A with respect to J over certain number field extension M of K. The notion of visible elements…