Sato-Tate groups of genus 2 curves

We consider the curves $ y^2=x^{2^m} -c$ and $y^2=x^{2^{d}+1}-cx$ over the rationals where $c \in \mathbb{Q}^{\times}.$ These curves are related via their associated Jacobian varieties in that the Jacobians of the latter appear as factors of the Jacobians of the former. One of the principle aims of this paper is to fully describe their Sato-Tate groups and distributions by determining generators of the component groups. In order to do this, we first prove the nondegeneracy of...

We determine the twisting Sato-Tate group of the genus $3$ hyperelliptic curve $y^2 = x^{8} - 14x^4 + 1$ and show that all possible subgroups of the twisting Sato-Tate group arise as the Sato-Tate group of an explicit twist of $y^2 = x^{8} - 14x^4 + 1$. Furthermore, we prove the generalized Sato-Tate conjecture for the Jacobians of all $\mathbb Q$-twists of the curve $y^2 = x^{8} - 14x^4 + 1$.

We consider the distribution of normalized Frobenius traces for two families of genus 3 hyperelliptic curves over Q that have large automorphism groups: y^2=x^8+c and y^2=x^7-cx with c in Q*. We give efficient algorithms to compute the trace of Frobenius for curves in these families at primes of good reduction. Using data generated by these algorithms, we obtain a heuristic description of the Sato-Tate groups that arise, both generically and for particular values of c. We the...

We announce the classification of Sato-Tate groups of abelian threefolds over number fields; there are 410 possible conjugacy classes of closed subgroups of USp(6) that occur. We summarize the key points of the "upper bound" aspect of the classification, and give a more rigorous treatment of the "lower bound" by realizing 33 groups that appear in the classification as maximal cases with respect to inclusions of finite index. Further details will be provided in a subsequent pa...

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 show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a...

The vertical Sato--Tate conjectures gives expected trace distributions for for families of curves. We develop exact expression for the distribution associated to degree-$4$ representations of $\mathrm{USp}(4)$, $\mathrm{SU}(2)\times\mathrm{SU}(2)$ and $\mathrm{SU}(2)$ in the neighborhood of the extremities of the Weil bound. As a consequence we derive qualitative distinctions between the extremal traces arising from generic genus-$2$ curves and genus-$2$ curves with real or q...

In this paper, we give a complete characterization of the component group of the Sato-Tate group of an abelian variety $A$ of arbitrary dimension, defined over a number field $K,$ in terms of the connectedness of the Lefschetz group associated to $A.$

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

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.

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 recent progress developed in the area.