Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

Edgar Costa; Ravi Donepudi; Ravi Fernando; Valentijn Karemaker; Caleb Springer; Mckenzie West

doi:10.1007/978-3-030-80914-0_7

Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

Edgar Costa^*, Ravi Donepudi, Ravi Fernando, Valentijn Karemaker, Caleb Springer, Mckenzie West

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

Abstract

Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

Original language	English
Title of host publication	Arithmetic Geometry, Number Theory, and Computation
Editors	Jennifer S. Balakrishnan, Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew Sutherland, John Voight
Publisher	Springer
Pages	259–276
Number of pages	15
Edition	1
ISBN (Electronic)	978-3-030-80914-0
ISBN (Print)	978-3-030-80913-3
DOIs	https://doi.org/10.1007/978-3-030-80914-0_7
Publication status	Published - 2022

Publication series

Name	Simons Symposia
Publisher	Springer
ISSN (Print)	2365-9564

Access to Document

10.1007/978-3-030-80914-0_7

978-3-030-80914-0_7Final published version, 361 KBLicence: Taverne

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Costa, E., Donepudi, R., Fernando, R., Karemaker, V., Springer, C., & West, M. (2022). Restrictions on Weil polynomials of Jacobians of hyperelliptic curves. In J. S. Balakrishnan, N. Elkies, B. Hassett, B. Poonen, A. Sutherland, & J. Voight (Eds.), Arithmetic Geometry, Number Theory, and Computation (1 ed., pp. 259–276). (Simons Symposia). Springer. https://doi.org/10.1007/978-3-030-80914-0_7

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title = "Restrictions on Weil polynomials of Jacobians of hyperelliptic curves",

abstract = " Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts. ",

author = "Edgar Costa and Ravi Donepudi and Ravi Fernando and Valentijn Karemaker and Caleb Springer and Mckenzie West",

note = "15 pages, 1 figure. To appear in the Simons Symposia proceedings volume for the Simons Collaboration {"}Arithmetic Geometry, Number Theory and Computation{"}",

year = "2022",

doi = "10.1007/978-3-030-80914-0_7",

language = "English",

isbn = "978-3-030-80913-3",

series = "Simons Symposia",

publisher = "Springer",

pages = "259–276",

editor = "Balakrishnan, {Jennifer S. } and Elkies, {Noam } and Hassett, {Brendan } and Poonen, {Bjorn } and Sutherland, {Andrew } and Voight, {John }",

booktitle = "Arithmetic Geometry, Number Theory, and Computation",

address = "Germany",

edition = "1",

}

Costa, E, Donepudi, R, Fernando, R, Karemaker, V, Springer, C & West, M 2022, Restrictions on Weil polynomials of Jacobians of hyperelliptic curves. in JS Balakrishnan, N Elkies, B Hassett, B Poonen, A Sutherland & J Voight (eds), Arithmetic Geometry, Number Theory, and Computation. 1 edn, Simons Symposia, Springer, pp. 259–276. https://doi.org/10.1007/978-3-030-80914-0_7

Restrictions on Weil polynomials of Jacobians of hyperelliptic curves. / Costa, Edgar; Donepudi, Ravi; Fernando, Ravi et al.
Arithmetic Geometry, Number Theory, and Computation. ed. / Jennifer S. Balakrishnan; Noam Elkies; Brendan Hassett; Bjorn Poonen; Andrew Sutherland; John Voight. 1. ed. Springer, 2022. p. 259–276 (Simons Symposia).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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T1 - Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

AU - Costa, Edgar

AU - Donepudi, Ravi

AU - Fernando, Ravi

AU - Karemaker, Valentijn

AU - Springer, Caleb

AU - West, Mckenzie

N1 - 15 pages, 1 figure. To appear in the Simons Symposia proceedings volume for the Simons Collaboration "Arithmetic Geometry, Number Theory and Computation"

PY - 2022

Y1 - 2022

N2 - Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

AB - Inspired by experimental data, this paper investigates which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed ${g\geq1}$, the proportion of isogeny classes of $g$ dimensional abelian varieties defined over $\mathbb{F}_q$ which fail this condition is $1 - Q(2g + 2)/2^g$ as $q\to\infty$ ranges over odd prime powers, where $Q(n)$ denotes the number of partitions of $n$ into odd parts.

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DO - 10.1007/978-3-030-80914-0_7

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EP - 276

BT - Arithmetic Geometry, Number Theory, and Computation

A2 - Balakrishnan, Jennifer S.

A2 - Elkies, Noam

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A2 - Poonen, Bjorn

A2 - Sutherland, Andrew

A2 - Voight, John

PB - Springer

ER -

Restrictions on Weil polynomials of Jacobians of hyperelliptic curves

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