From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups

Karabulut, Cihan

doi:10.1016/j.jnt.2021.07.010

Journal of Number Theory

2022

DOI: 10.1016/j.jnt.2021.07.010

|View full text |Cite

From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups

Cihan Karabulut

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Bianchi period polynomials: Hecke action and congruences

Combes

2024

Res. number theory

View full text Add to dashboard Cite

Let $$\Gamma ={{\,\mathrm{\textrm{PSL}}\,}}_2(\mathcal {O})$$ Γ = PSL 2 ( O ) be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for $$\Gamma $$ Γ is “dual” to the space of weight k modular symbols for $$\Gamma $$ Γ , reflecting the duality between the first and second cohomology groups of the arithmetic group $$\Gamma $$ Γ . Using this result, we describe the action of Hecke operators on the space of period polynomials for $$\Gamma $$ Γ via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

show abstract

Bianchi period polynomials: Hecke action and congruences

Combes

2024

Res. number theory

View full text Add to dashboard Cite

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

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

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.

From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups

Cited by 1 publication

References 17 publications

Bianchi period polynomials: Hecke action and congruences

Bianchi period polynomials: Hecke action and congruences

Contact Info

Product

Resources

About