Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$

Emory, Melissa; Goodson, Heidi

doi:10.48550/arxiv.2004.10583

Cited by 2 publications

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“…The Jacobian of C p,q is absolutely simple and nondegenerate (see Proposition 2.2), and so, as in [9,10], we can go one step further and conclude that the containment is actually an equality. It follows that we can choose ST 0 (Jac(C p,q )) to be the maximal compact subgroup U(1) g .…”

Section: 2mentioning

confidence: 85%

“…As noted in the Introduction, this lemma can be applied to more varieties than those we study in this paper. For example, this lemma could be used for the varieties studied in [9,10,23] and [15] (for certain values of c).…”

Section: Endomorphisms Of Catalan Jacobiansmentioning

confidence: 99%

“…As noted in [16], it is not expected, in dimension greater than 3, that every group satisfying the Sato-Tate axioms (see Section 2.4) can be realized using Jacobians of curves. There are currently two articles that study families of Jacobian varieties of arbitrarily high genus (see [9,10]). In this article we provide a new example of an infinite family of Sato-Tate groups that can be realized by higher dimensional Jacobian varieties.…”

Section: Introductionmentioning

confidence: 99%

“…For any nondegenerate abelian variety with endomorphism given by a diagonal matrix and with CM by a cyclotomic field, this lemma can be used to determine the component group generators. This may seem like a very specific scenario, but we note that this describes the Jacobians considered in, for example, [9,10,23] and [15] (for certain values of c).…”

Section: Introductionmentioning

confidence: 99%

“…We begin by defining both the algebraic Sato-Tate group and the Sato-Tate group. We follow the exposition of [9] and [31,Section 3.2]. See also [29,Chapter 8].…”

Section: Introductionmentioning

confidence: 99%

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Sato-Tate Distributions of Catalan Curves

Goodson¹

2021

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For distinct odd primes p and q, we define the Catalan curve Cp,q by the affine equation y q = x p − 1. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting distributions of coefficients of their normalized L-polynomials. Catalan Jacobians are nondegenerate and simple with noncyclic Galois groups (of the endomorphism fields over Q), thus making them interesting varieties to study in the context of Sato-Tate groups. We compute both statistical and numerical moments for the limiting distributions. Lastly, we determine the Galois endomorphism types of the Jacobians using both old and new techniques.

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Section: 2mentioning

confidence: 85%

Section: Endomorphisms Of Catalan Jacobiansmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We begin by defining both the algebraic Sato-Tate group and the Sato-Tate group. We follow the exposition of [9] and [31,Section 3.2]. See also [29,Chapter 8].…”

Section: Introductionmentioning

confidence: 99%

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Sato-Tate Distributions of Catalan Curves

Goodson¹

2021

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Determining monodromy groups of abelian varieties

Zywina¹

2020

Preprint

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Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, ℓ-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.Corollary 1.3. Assume that Conjectures 4.3 and 4.5 hold for A. Let S be the (finite) set of primes ℓ for which (G • A ) Q ℓ is not quasi-split. Then for almost all prime ideals q and p of O K that split completely in K conn A , the polynomials P A,q (x) and P A,p (x) determine the group G • A,ℓ and the representation V ℓ (A) of G • A,ℓ up to isomorphism for all ℓ / ∈ S.Proof. From Theorem 1.1, we may assume that we know the root datum Ψ(G • A ), the homomorphismA , and the weights of G • A ⊆ GL V A with multiplicities. Take any prime ℓ / ∈ S and set, where the equality uses the Mumford-Tate conjecture assumption. By choosing an embedding Q ֒→ Q ℓ , we obtain the root datum Ψ(G) and the associated homomorphism µ G : Gal(Q ℓ /Q ℓ ) → Out(Ψ(G)). Since G is quasi-split, this information determines the algebraic group G = G • A,ℓ up to isomorphism, cf. §2.9. Since we know the weights of the representation (Gwith multiplicities, we also have enough information to determine the representation V ℓ (A) of G • A,ℓ up to isomorphism. Finally, the set S is finite, cf. Theorem 6.7 of [PR94].

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Sato-Tate Distributions of $y^2=x^p-1$ and $y^2=x^{2p}-1$

Cited by 2 publications

References 0 publications

Sato-Tate Distributions of Catalan Curves

Sato-Tate Distributions of Catalan Curves

Determining monodromy groups of abelian varieties

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