Related papers: Sato-Tate Distributions
We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual…
In this expository note, we present an approach to the generalization of Serre of the Sato-Tate Conjecture. Most of its content is taken from Serre's original references. However, we provide a few new examples and supply references to…
We describe the analogue of the Sato-Tate conjecture for an abelian variety over a number field; this predicts that the zeta functions of the reductions over various finite fields, when properly normalized, have a limiting distribution…
For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random…
Given an abelian variety over a number field, its Sato-Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L-function of the abelian variety. It was previously shown by Fit\'e, Kedlaya,…
Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture,…
We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of $\GL_2$-type and defined over a totally real number…
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image…
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…
Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any…
We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of…
Let $A$ be an abelian variety defined over a number field and let $G$ denote its Sato-Tate group. Under the assumption of certain standard conjectures on $L$-functions attached to the irreducible representations of $G$, we study the…
We prove a few new cases of the Sato-Tate conjecture for abelian surfaces, using a new automorphy theorem of Allen et al. Then in the unproven cases, we use partial results to describe nontrivial asymptotics on the trace of Frobenius, and…
In this paper, we state a hybrid Chebotarev-Sato-Tate conjecture for abelian varieties and we prove it in several particular cases using current potential automorphy theorems.
This survey describe Hodge, Tate and Mumford-Tate conjectures for abelian varieties. After some preliminaries on endomorphism ring, polarization and algebraic cycles, we state the three conjectures and provide a list of know results.…
From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we…
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize…
We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at…
We give some arithmetic-geometric interpretations of the moments M_2[a_1], M_1[a_2], and M_1[s_2] of the Sato-Tate group of an abelian variety A defined over a number field by relating them to the ranks of the endomorphism ring and…