Sato–Tate distributions and Galois endomorphism modules in genus 2
@article{Fite2012SatoTateDA, title={Sato–Tate distributions and Galois endomorphism modules in genus 2}, author={Francesc Fit'e and Kiran S. Kedlaya and V{\'i}ctor Rotger and Andrew V. Sutherland}, journal={Compositio Mathematica}, year={2012}, volume={148}, pages={1390 - 1442} }
Abstract 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 matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according…
82 Citations
Sato-Tate groups of genus 2 curves
- MathematicsAdvances on Superelliptic Curves and their Applications
- 2015
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…
Sato-Tate groups of abelian threefolds
- Mathematics
- 2019
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…
An algebraic Sato-Tate group and Sato-Tate conjecture
- Mathematics
- 2011
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…
SATO-TATE DISTRIBUTIONS
- Mathematics
- 2018
In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the…
Effective Sato-Tate conjecture for abelian varieties and applications
- Mathematics
- 2020
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 Distributions of Catalan Curves
- Mathematics
- 2021
For distinct odd primes p and q, we define the Catalan curve Cp,q by the affine equation y = x − 1. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting…
Determining monodromy groups of abelian varieties
- Mathematics
- 2020
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…
Sato-Tate distributions of twists of y^2=x^5-x and y^2=x^6+1
- Mathematics
- 2014
We determine the limiting distribution of the normalized Euler factors of an abelian surface A defined over a number field k when A is isogenous to the square of an elliptic curve defined over k with…
On the Rank and the Convergence Rate Toward the Sato–Tate Measure
- Mathematics
- 2017
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…
COMBINATORIAL INTERPRETATIONS OF THE TRACE MOMENT SEQUENCES OF SUBGROUPS OF USp(4)
- Mathematics
- 2014
Fit e et al. (2011) describe a generalization of the Sato-Tate conjecture by hypothe- sizing that the distribution as p varies, for a xed algebraic curve, of the normalized error terms ((p + 1 number…
References
SHOWING 1-10 OF 65 REFERENCES
An algebraic Sato-Tate group and Sato-Tate conjecture
- Mathematics
- 2011
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…
HYPERELLIPTIC JACOBIANS WITHOUT COMPLEX MULTIPLICATION
- Mathematics
- 1999
has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] is “very big”. More precisely, if f is a polynomial of degree n ≥…
Modular Curves and Abelian Varieties
- Mathematics
- 2012
Stable reduction of modular curves.- On p-adic families of automorphic forms.- ?-curves and abelian varieties of GL2-type from dihedral genus 2 curves.- The old subvariety of J0(NM).- Irreducibility…
Hyperelliptic curves, L-polynomials, and random matrices
- Mathematics
- 2008
We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement…
Complex multiplication of abelian surfaces
- Mathematics, Computer Science
- 2010
The main theme of this thesis is making these constructions explicit for the case where the abelian varieties have dimension 2, and gives an algorithm for computing class polynomials for quartic CM-fields, based on an algorithm of Spallek.
The Sato-Tate conjecture for Hilbert modular forms
- Mathematics
- 2009
We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations…
Abelian L-adic representation and elliptic curves
- MathematicsAdvanced book classics
- 1989
This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent…
Order computations in generic groups
- Mathematics, Computer Science
- 2007
It is proved that a generic algorithm can compute |α| for all α ∈ S ⊆ G in near linear time plus the cost of a single order computation with N = λ(S), and it is shown that in most cases the structure of an abelian group G can be determined using an additional O (Nδ/4 ) group operations, given an O ( Nδ ) bound on |G| = N.