Study and computation of a Hurwitz space and totally real <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="italic">PSL</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="double-struck">F</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math>-extensions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:mi

Hallouin, Emmanuel

doi:10.1016/j.jalgebra.2005.04.026

Journal of Algebra

2005

DOI: 10.1016/j.jalgebra.2005.04.026

|View full text |Cite

Study and computation of a Hurwitz space and totally real PSL2(F8)-extensions of

Emmanuel Hallouin¹

Abstract: Developing on works by Fried, Völklein, Matzat, Malle, Dèbes, Wewers, we give a method for computing a Hurwitz space and illustrate it on some example of number theoretic interest: we study and compute a family of degree 9 covers of P 1 C with monodromy group PSL 2 (F 8 ) and having four branch points. We deduce explicit regular PSL 2 (F 8 )-extensions of the rational function field Q(ϕ) with totally real fibers. This gives rise to totally real polynomials over Q with Galois group PSL 2 (F 8 ).

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2009

2020

Publication Types

Select...

Article4

Relationship

Self Cite1

Independent3

Authors

Journals

Cited by 4 publications

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Computation of a cover of Shimura curves using a Hurwitz space

Hallouin¹

2009

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Using a Hurwitz space computation, we determine the canonical model of the cover of Shimura curves X 0 (2) → X (1) associated to the quaternion algebra over the cubic field of discriminant 13 2 , which is ramified at exactly two real places and unramified at finite places. Then, we list the coordinates of some rational CM points on X (1). IntroductionIn his classic "Shimura curve computations" [Elk98], Elkies considers Shimura curves from a computational point of view. He explicitly determines covers between Shimura curves associated to the same quaternion algebra and also coordinates of some CM points of those curves. His approach relies mostly on the uniformization of these curves by a quotient of the hyperbolic plane. Several mathematicians have investigated this kind of computational challenge [Voi06,BT07,GR04]. Elkies himself published a second article on the area [Elk06], where he computes the canonical models of some Shimura curves associated with quaternion algebras whose center is one of the cubic cyclic totally real fields of discriminant 49 = 7 2 or 81 = 9 2 and ramified at no finite place and exactly two of the three real places of the center. He ends his article by describing the cover of Shimura curves X 0 (2) → X (1) (see Section 1 for definitions) for the cubic field K of discriminant 13 2 . More precisely, he shows that the curve X (1) has genus zero and that the cover X 0 (2) → X (1) is a degree 9 cover with a certain ramification behavior. But he does not compute equations for the canonical model of this cover. In this work, we compute such a model explicitly.We also illustrate the utility of the computation of Hurwitz spaces. A great tool to compute an explicit model of a cover of P 1 C with given ramification data is to consider the whole family of covers

show abstract

Computation of a cover of Shimura curves using a Hurwitz space

Hallouin¹

2009

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

Barth

König

Wenz

2020

Journal of Symbolic Computation

View full text Add to dashboard Cite

We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of 3-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp 6 (2).Note: Supplementary data are contained in an extra file, available at:https://arxiv.org/src/1803.08778/anc/anc.txt

show abstract

Hurwitz–Belyi maps

Roberts¹

2018

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

View full text Add to dashboard Cite

The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain onedimensional slices of these coverings are particularly interesting Belyi maps. We present systematic examples of such "Hurwitz-Belyi maps." Our examples illustrate a wide variety of theoretical phenomena and computational techniques. arXiv:1608.08302v1 [math.NT] 30 Aug 2016 2 DAVID P. ROBERTS

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Study and computation of a Hurwitz space and totally real PSL2(F8)-extensions of

Cited by 4 publications

References 7 publications

Computation of a cover of Shimura curves using a Hurwitz space

Computation of a cover of Shimura curves using a Hurwitz space

An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

Hurwitz–Belyi maps

Contact Info

Product

Resources

About