Sato–Tate distributions of twists of the Fermat and the Klein quartics

Fité, Francesc; García, Elisa Lorenzo; Sutherland, Andrew V.

doi:10.1007/s40687-018-0162-0

Cited by 12 publications

(16 citation statements)

References 27 publications

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“…• In dimension 3, the joint coefficient distribution of the normalized L-polynomials is not determined by the individual coefficient distributions of the normalized Lpolynomials, as is the case in dimension 2. This was already reported in [FLS18].…”

supporting

confidence: 75%

“…• In dimension 3, one finds component groups that are not solvable, namely the simple group of order 168 and its double cover. These appeared previously in the analysis of Sato-Tate groups of twists of the Klein quartic [FLS18]. • One also finds for the first time component groups which are too big to be explained by twisting curves, notably the Hessian group of order 216 (which exceeds the Hurwitz bound of 168 for the automorphism group of a genus 3 curve) and its double cover.…”

mentioning

confidence: 69%

“…It is also useful to consider twists of curves with extremal automorphism groups. In lieu of analyzing such cases here, we will draw on prior results from [ACLLM18], [FLS18], [FS16].…”

Section: Methods For Explicit Realizations Of Abelian Threefoldsmentioning

confidence: 99%

“…As above, the total time for B = 2 30 is one or two core-weeks. • For Jacobians A of twists of the Fermat and Klein quartics X 4 + Y 4 + Z 4 = 0 and X 3 Y + Y 3 Z + XZ 3 = 0, both covers of genus 1 curves, we can compute L p (T ) modulo p as above and apply the method described in [FLS18] to lift L p (T ) to Z[T ], which takes (log p) 2+o(1) time. Alternatively (and more efficiently), we can compute L p (T ) using an isogeny twist of E 3 by a 3-dimensional Artin representation ρ M , where E/Q is an elliptic curve with CM by M whose cube is Q-isogenous to A.…”

Section: Methods For Explicit Realizations Of Abelian Threefoldsmentioning

confidence: 99%

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Sato-Tate groups of abelian threefolds

Fité¹,

Kedlaya²,

Sutherland³

2021

Preprint

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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 Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato-Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato-Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in GAP, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L-functions.

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supporting

confidence: 75%

mentioning

confidence: 69%

“…It is also useful to consider twists of curves with extremal automorphism groups. In lieu of analyzing such cases here, we will draw on prior results from [ACLLM18], [FLS18], [FS16].…”

Section: Methods For Explicit Realizations Of Abelian Threefoldsmentioning

confidence: 99%

Section: Methods For Explicit Realizations Of Abelian Threefoldsmentioning

confidence: 99%

See 2 more Smart Citations

Sato-Tate groups of abelian threefolds

Fité¹,

Kedlaya²,

Sutherland³

2021

Preprint

Self Cite

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“…For example, [11] and [12,16] determine all possible Sato-Tate groups in dimension 2 and 3, respectively, by determining which subgroups of the unitary symplectic group satisfy certain axioms (see Section 2.4). Other articles determine Sato-Tate groups for certain families of genus 2 and 3 curves (see [15,23]) or for twists of curves (see [3,13,14]).…”

Section: Introductionmentioning

confidence: 99%

Sato-Tate Distributions of Catalan Curves

Goodson¹

2021

Preprint

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

Fité

Guitart

2021

Math. Z.

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We introduce a tensor decomposition of the -adic Tate module of an abelian variety A 0 defined over a number field which is geometrically isotypic and potentially of GL 2 -type. We use this decomposition as a fundamental tool to describe the Sato-Tate group of A 0 and to prove the Sato-Tate conjecture in certain cases. Contents1. Introduction 1 Conventions and notations. 4 Acknowledgements 4 2. A Tate module tensor decomposition 4 3. The weakly compatible system V λ (B) 10 4. Sato-Tate groups and Sato-Tate conjecture 11 5. Scenarios of applicability 17 Examples: modular abelian varieties 19 References 20

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Sato–Tate distributions of twists of the Fermat and the Klein quartics

Cited by 12 publications

References 27 publications

Sato-Tate groups of abelian threefolds

Sato-Tate groups of abelian threefolds

Sato-Tate Distributions of Catalan Curves

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

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