Sato–Tate Distributions of Catalan Curves

Heidi Goodson

doi:10.5802/jtnb.1238

Heidi Goodson ¹

¹ Department of Mathematics Brooklyn College, City University of New York 2900 Bedford Avenue, Brooklyn, NY 11210 USA

Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 87-113.

Abstract
Résumé

For distinct odd primes $p$ and $q$ , we define the Catalan curve $C_{p, q}$ 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.

Étant donnés deux nombres premiers impairs distincts $p$ et $q$ , nous définissons la courbe de Catalan $C_{p, q}$ donnée par l’équation affine $y^{q} = x^{p} - 1$ . Dans cet article, nous construisons les groupes de Sato–Tate des variétés jacobiennes de ces courbes, afin d’étudier les distributions asymptotiques des coefficients de leurs polynômes de Weil normalisés. Ces jacobiennes de Catalan sont non-dégénérées et simples et le groupe de Galois de leur corps d’endomorphismes sur $ℚ$ n’est pas cyclique, ce qui en font des variétés intéressantes dans le contexte des groupes de Sato–Tate. Dans cet article, nous calculons les moments statistiques et numériques des distributions asymptotiques. Enfin, nous déterminons les types des modules galoisiens donnés par l’algèbre réelle des endomorphismes de ces jacobiennes, en utilisant des techniques connues ainsi que certaines nouvelles techniques.

Received: 2021-10-21
Revised: 2022-10-04
Accepted: 2022-10-21
Published online: 2023-05-04

DOI: 10.5802/jtnb.1238

Classification: 11M50, 11G10, 11G20, 14G10
Keywords: Sato–Tate groups, Sato–Tate distributions, Jacobian varieties, endomorphism algebras

Author's affiliations:

Heidi Goodson ¹

¹ Department of Mathematics Brooklyn College, City University of New York 2900 Bedford Avenue, Brooklyn, NY 11210 USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Sato{\textendash}Tate {Distributions} of {Catalan} {Curves}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Heidi Goodson. Sato–Tate Distributions of Catalan Curves. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 87-113. doi : 10.5802/jtnb.1238. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1238/

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