Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Experiment Videos

Adjoint modular Galois representations and their Selmer groups.

H Hida1, J Tilouine, E Urban

  • 1Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA.

Proceedings of the National Academy of Sciences of the United States of America
|October 19, 2001
PubMed
Summary

This study presents class number formulas for Selmer groups of modular Galois representations. It connects these formulas to L-functions and Iwasawa theory, offering new insights into number theory conjectures.

Related Concept Videos

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Indian agriculture, air pollution, and public health in the age of COVID.

World development·2020
Same author

[Reevaluating the shelf life of sterilized packaged items via a risk-analysis study].

Annales pharmaceutiques francaises·2020
Same author

Impact of EUCAST ceftaroline breakpoint change on the susceptibility of methicillin-resistant Staphylococcus aureus isolates collected from patients with complicated skin and soft-tissue infections.

Clinical microbiology and infection : the official publication of the European Society of Clinical Microbiology and Infectious Diseases·2019
Same author

Characterization of Pasteurella multocida strains isolated from human infections.

Comparative immunology, microbiology and infectious diseases·2019
Same author

An overview of the data obtained during the validation of an optimized MALDI-TOF MS Biotyper database for the identification of anaerobic bacteria.

Data in brief·2018
Same author

Validation of MALDI-TOF MS Biotyper database optimized for anaerobic bacteria: The ENRIA project.

Anaerobe·2018

Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Representation Theory

Background:

  • Recent advancements in proving class number formulas and main conjectures.
  • Focus on Selmer groups of the adjoint representation of 2D modular Galois representations.
  • Building upon the p-adic Galois representation of modular elliptic curves.

Purpose of the Study:

  • To derive formulas for Selmer groups related to modular Galois representations.
  • To generalize these formulas within an Iwasawa theoretic framework.
  • To state and provide evidence for a main conjecture in a two-variable Iwasawa setting.

Main Methods:

  • Expressing intersection numbers using L-functions.
  • Deducing Selmer group orders from the Shimura-Taniyama conjecture.

Related Experiment Videos

  • Generalizing to one and two-variable Iwasawa theory using Hecke algebras and cyclotomic characters.
  • Utilizing p-adic L-functions and p-adic Siegel modular forms.

    • Main Results:

    • A formula relating Selmer group orders to L(1, ad(phi0)).
    • A one-variable Iwasawa formula for the characteristic power series of Selmer groups.
    • A statement of the main conjecture for the two-variable case involving Selmer groups and cyclotomic characters.

    Conclusions:

    • The study establishes key formulas and conjectures in the arithmetic of modular forms and Galois representations.
    • It provides a framework for understanding Selmer groups through p-adic L-functions and Iwasawa theory.
    • Recent results and strategies for proving the main conjecture using p-adic modular forms are outlined.