Sato-Tate Group Research Articles

An Exploration of Degeneracy in Abelian Varieties of Fermat Type

The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles – Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form y 2 = x m − 1 . We prove that the Jacobian of the curve is degenerate whenever m is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.

Effective Sato–Tate conjecture for abelian varieties and applications

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 give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A . We discuss three applications of this conditional result. First, for an abelian variety defined over k , we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius traces attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A' defined over k with no common factors up to k -isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A' have opposite sign.

Determining monodromy groups of abelian varieties

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 practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.

Sato-Tate distributions of y2 = xp − 1 and y2 = x2p − 1

We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y2=xp−1 and y2=x2p−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves.

Machine-learning the Sato–Tate conjecture

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently.

Tate module tensor decompositions and the Sato–Tate conjecture for certain abelian varieties potentially of $$\mathrm {GL}_2$$-type

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 $$\mathrm {GL}_2$$ -type and defined over a totally real number field, we use this decomposition to describe its Sato–Tate group and to prove the Sato–Tate conjecture in certain cases.

Towards the Sato–Tate groups of trinomial hyperelliptic curves

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form [Formula: see text] where [Formula: see text] is the genus of the curve and [Formula: see text] is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for [Formula: see text] curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus [Formula: see text] curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components [Formula: see text] for these families of curves.

Arithmetic invariants from Sato–Tate moments

We give some arithmetic-geometric interpretations of the moments M2[a1], M1[a2], and M1[s2] 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 Néron–Severi group of A.

The twisting Sato–Tate group of the curve $$y^2 = x^{8} - 14x^4 + 1$$ y 2 = x 8 - 14 x 4 + 1

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 $$y^2 = x^{8} - 14x^4 + 1$$ .

Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato–Tate groups over a number field

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that is isogenous over $\overline{\mathbb{Q}}$ to $E^g$, where $E$ is an elliptic curve. If $E$ does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic $\mathbb{Q}$-curves $E$ is isogenous to a curve defined over a polyquadratic extension of $\mathbb{Q}$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato--Tate groups: First, we show that $18$ of the $34$ possible Sato--Tate groups of abelian surfaces over $\mathbb{Q}$ occur among at most $51$ $\overline{\mathbb{Q}}$-isogeny classes of abelian surfaces over $\mathbb{Q}$; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the $52$ possible Sato--Tate groups of abelian surfaces.

On the Rank and the Convergence Rate Toward the Sato–Tate Measure

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 convergence rate of any virtual character of G. We find that this convergence rate is dictated by several arithmetic invariants of A, such as its rank or its Sato–Tate group G. The results are consonant with some previous experimental observations, and we also provide additional numerical evidence consistent with them. The techniques that we use were introduced by Sarnak in a letter to Mazur, in order to explain the bias in the sign of the Frobenius traces of an elliptic curve without complex multiplication defined over Q. We show that the same methods can be adapted to study the convergence rate of the characters of its Sato–Tate group, and that they can also be employed in the more general case of abelian varieties over number fields. A key tool in our analysis is the existence of limiting distributions for automorphic L-functions, which is due to Akbary, Ng, and Shahabi.

Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent

AbstractLetdenote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() ofsplits over ℚ as the product of Jacobians Jac(k), 1 ≤k≤ ℓ −2, wherekare curves obtained as quotients ofby certain subgroups of automorphisms of. It is well known that Jac(k) is the power of an absolutely simple abelian varietyBkwith complex multiplication. We call degenerate those pairs (ℓ,k) for whichBkhas degenerate CM type. For a non-degenerate pair (ℓ,k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether (ℓ,k) is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the ℓ-th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

Character theory approach to Sato–Tate groups

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points provide satisfactory convergence. Even for $g=2$, the classical approach using moment statistics requires about $2^{30}$ sample points to obtain such information.

An algebraic Sato-Tate group and Sato-Tate conjecture

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 Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to the Mumford-Tate group, but the group of components carries additional arithmetic information. We then check that in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fite, Kedlaya, Rotger, and Sutherland.

Sato–Tate distributions of twists ofy2=x5−xandy2=x6+ 1

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 complex multiplication. As an application, we prove the Sato-Tate Conjecture for Jacobians of Q-twists of the curves y^2=x^5-x and y^2=x^6+1, which give rise to 18 of the 34 possibilities for the Sato-Tate group of an abelian surface defined over Q. With twists of these two curves one encounters, in fact, all of the 18 possibilities for the Sato-Tate group of an abelian surface that is isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato-Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato-Tate group of its Jacobian.

Sato–Tate distributions and Galois endomorphism modules in genus 2

AbstractFor 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 to the Galois module structure on the ℝ-algebra generated by endomorphisms of $A_{{\overline {\mathbb Q}}}$ (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.