Sato-Tate groups of abelian threefolds

Fité, Francesc; Kedlaya, Kiran S.; Sutherland, Andrew V.

doi:10.48550/arxiv.2106.13759

Cited by 3 publications

(3 citation statements)

References 60 publications

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“…The next two results represent partial progress toward Theorem 1.1 and will be used to simplify some proofs in subsequent sections. We use the notations for Sato-Tate groups introduced in [6] and [8], and present in the data base [14].…”

Section: Consequences Of the Classification Of Sato-tate Groupsmentioning

confidence: 99%

On a local-global principle for quadratic twists of abelian varieties

Fité

2022

Math. Ann.

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Let A and $$A'$$ A ′ be abelian varieties defined over a number field k of dimension $$g\ge 1$$ g ≥ 1 . For $$g\le 3$$ g ≤ 3 , we show that the following local-global principle holds: A and $$A'$$ A ′ are quadratic twists of each other if and only if, for almost all primes $$\mathfrak {p}$$ p of k of good reduction for A and $$A'$$ A ′ , the reductions $$A_\mathfrak {p}$$ A p and $$A_\mathfrak {p}'$$ A p ′ are quadratic twists of each other. This result is known when $$g=1$$ g = 1 , in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension $$g=4$$ g = 4 .

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Section: Consequences Of the Classification Of Sato-tate Groupsmentioning

confidence: 99%

On a local-global principle for quadratic twists of abelian varieties

Fité

2022

Math. Ann.

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“…They will be used to simplify some proofs in subsequent sections. We use the notations for Sato-Tate groups introduced in [FKRS12] and [FKS21c], and present in the L-functions and modular forms data base [LMFDB].…”

Section: Consequences Of the Classification Of Sato-tate Groupsmentioning

confidence: 99%

On a local-global principle for quadratic twists of abelian varieties

Fité¹

2021

Preprint

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Let A and A ′ be abelian varieties defined over a number field k of dimension g ≥ 1. For g ≤ 3, we show that the following local-global principle holds: A and A ′ are quadratic twists of each other if and only if, for almost all primes p of k of good reduction for A and A ′ , the reductions Ap and A ′ p are quadratic twists of each other. This result is known when g = 1, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension g = 4.

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“…Nevertheless, for general genus 3 curve the algorithms we present substantially extend the practical range of N one may consider. This played a key role in [11,12] where a preliminary version of our algorithm was used to compute Sato-Tate distributions, and in computing the L-functions of the nonhyperelliptic genus 3 curves tabulated in [34].…”

Section: Introductionmentioning

confidence: 99%

Counting points on smooth plane quartics

2022

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We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p. We address both the case where X is defined over $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O(N) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest.

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Sato-Tate groups of abelian threefolds

Cited by 3 publications

References 60 publications

On a local-global principle for quadratic twists of abelian varieties

On a local-global principle for quadratic twists of abelian varieties

On a local-global principle for quadratic twists of abelian varieties

Counting points on smooth plane quartics

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