## Examples of CM curves of genus two defined over the reflex field

Research output: Contribution to journalArticle

### Standard

In: LMS Journal of Computation and Mathematics, 01.08.2015.

Research output: Contribution to journalArticle

### Author

Bouyer, Florian ; Streng, Marco. / Examples of CM curves of genus two defined over the reflex field. In: LMS Journal of Computation and Mathematics. 2015.

### Bibtex

@article{739d55085588460abb17e39a81198979,
title = "Examples of CM curves of genus two defined over the reflex field",
abstract = "In {"}Proving that a genus 2 curve has complex multiplication{"}, van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not feature in this list, as the field of definition in that case always contains a real quadratic field, known as the real quadratic subfield of the reflex field. We extend van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest {"}generic{"} examples of CM curves of genus two. We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.",
keywords = "math.NT, 11G15, 14K22",
author = "Florian Bouyer and Marco Streng",
note = "31 pages; Updated some references",
year = "2015",
month = "8",
day = "1",
doi = "10.1112/S1461157015000121",
language = "Undefined/Unknown",
journal = "LMS Journal of Computation and Mathematics",
issn = "1461-1570",
publisher = "London Mathematical Society",

}

### RIS - suitable for import to EndNote

TY - JOUR

T1 - Examples of CM curves of genus two defined over the reflex field

AU - Bouyer, Florian

AU - Streng, Marco

N1 - 31 pages; Updated some references

PY - 2015/8/1

Y1 - 2015/8/1

N2 - In "Proving that a genus 2 curve has complex multiplication", van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not feature in this list, as the field of definition in that case always contains a real quadratic field, known as the real quadratic subfield of the reflex field. We extend van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest "generic" examples of CM curves of genus two. We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

AB - In "Proving that a genus 2 curve has complex multiplication", van Wamelen lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). For each of the 19 curves, the CM-field turns out to be cyclic Galois over $\mathbf{Q}$. The generic case of non-Galois quartic CM-fields did not feature in this list, as the field of definition in that case always contains a real quadratic field, known as the real quadratic subfield of the reflex field. We extend van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest "generic" examples of CM curves of genus two. We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

KW - math.NT

KW - 11G15, 14K22

U2 - 10.1112/S1461157015000121

DO - 10.1112/S1461157015000121

M3 - Article

JO - LMS Journal of Computation and Mathematics

JF - LMS Journal of Computation and Mathematics

SN - 1461-1570

ER -